Every K₄-Minor-Free Graph Is Ramsey Size-Linear

1. Introduction

For graphs G and H, the Ramsey number $R(G,H)$ is the smallest integer N such that for every graph $\Gamma$ on N vertices, $\Gamma$ contains a copy of G or its complement $\overline{\Gamma }$ contains a copy of H. Harary conjectured that $R(K_3,H)\le 2e\left(H\right)+1$ for every graph H without isolated vertices; this was proved by Sidorenko [18] and by Goddard and Kleitman [14]. Erdős, Faudree, Rousseau and Schelp [12] initiated a systematic study of this phenomenon and called a graph GRamsey size-linear if there is a constant $C_G$ such that

$$R(G,H)\le C_{G}e\left(H\right)\mathrm{for}\mathrm{every}\mathrm{graph}H\mathrm{without}\mathrm{isolated}\mathrm{vertices}.$$

Among other results, they proved the following.

(i)

Every connected graph G with $e\left(G\right)\le v\left(G\right)+1$ is Ramsey size-linear [12, Theorem 4].

(ii)

If G contains a subgraph with k vertices and at least $2k-2$ edges, then G is not Ramsey size-linear [12, Corollary 1].

(iii)

If T is a tree, then $K_1+T$ is Ramsey size-linear; in fact $R(K_1+T,H)\le 2v\left(T\right)e\left(H\right)$ [12, Theorem 3 and Corollary 2].

(iv)

Every graph with Turán number $O\left(n^{3/2}\right)$ is Ramsey size-linear [12, Theorem 5]; examples mentioned there include $K_{3,3}-e$ and $Q_3-e$.

Fact (ii) comes from the lower bound $R(G,K_n)=\Omega \left({(n/logn)}^{m_2\left(G\right)}\right)$, obtained by the probabilistic method [19] (see also [12,20]), where $m_2\left(G\right)=max\{(e\left(F\right)-1)/(v\left(F\right)-2):F\subseteq G,v\left(F\right)\ge 3\}$: a subgraph with k vertices and $2k-2$ edges forces $m_2\left(G\right)>2$, and then $R(G,K_n)$ grows faster than $e\left(K_n\right)$. Erdős et al. asked whether this is the only obstruction.

Question 1.1

([12, Question 1]; Erdős Problem #566 [3]). Let G be a graph such that every subgraph of G on $k\ge 2$ vertices has at most $2k-3$ edges, that is, $m_2\left(G\right)\le 2$. Is G Ramsey size-linear?

They also asked whether $K_{3,3}$, the cube $Q_3$ and $K_4^{*}=K_5-(K_{1,2}\cup K_2)$, the graph obtained from $K_4$ by subdividing one edge, are Ramsey size-linear [12, Question 2], noting that each of them would be minimally non-Ramsey size-linear otherwise; see also [11], and [2,5] for $K_4^{*}$. Here a graph is minimally non-Ramsey size-linear if it is not Ramsey size-linear but all its proper subgraphs are. Erdős et al. observed that $K_4$ is such a graph and asked for others [12, Question 6] (Erdős Problem #79). Related questions are collected as Erdős Problems #566–#570 [3].

Facts (ii) and (iii) show that the density $2k-3$ in Question 1.1 cannot be raised and is attained by Ramsey size-linear graphs. Balister, Schelp and Simonovits [2] proved that joining two vertices at distance d of a Ramsey size-linear graph by a new path of length at least $d+3$ preserves the property. They asked whether every 2-connected graph of minimum degree at least three fails to be Ramsey size-linear; this was answered negatively by combining (iv) with a construction of Janzer [16], as observed in [5]. Bradač, Gishboliner and Sudakov [5] proved that every subdivision of $K_4$ with at least six vertices is Ramsey size-linear, that $R(K_4^{*},F)=O\left(e\left(F\right)\right)$ for every bipartite F, and that $R(K_4^{*},K_n)=O\left(n^{5/2}\right)$, and remarked that Question 1.1 seems to be out of reach. Wigderson [20, Open problem 5] proved that there are infinitely many minimally non-Ramsey size-linear graphs; his argument is non-constructive, and he asked for an explicit example other than $K_4$ [20]. For cycles, the optimal linear constant was recently determined by Cambie, Freschi, Morawski, Petrova and Pokrovskiy [7]; see also [15].

Main Result

Theorem 1.2.

Let G be a graph with no $K_4$-minor, equivalently a graph of treewidth at most two, and let $h=max\left\{v\right(G),3\}$. Then for every graph H with $m\ge 1$ edges and no isolated vertices,

$$R(G,H)\le (624h-1816)m.$$

In particular, every $K_4$-minor-free graph is Ramsey size-linear.

A graph of treewidth at most two on $k\ge 2$ vertices has at most $2k-3$ edges, with equality for 2-trees, and its subgraphs again have treewidth at most two. Thus Theorem 1.2 answers Question 1.1 affirmatively for a minor-closed class which contains graphs of the extremal density $2k-3$ allowed there, including all 2-trees, all outerplanar graphs and all series–parallel graphs. It extends facts (i) and (iii): the graphs $K_1+T$ have treewidth two, and a connected graph containing a subdivision of $K_4$ has cyclomatic number $e-v+1\ge 3$, so connected graphs with at most $v\left(G\right)+1$ edges have no $K_4$-minor. The smallest graphs covered by Theorem 1.2 but by none of the results quoted above have six vertices: the 3-sun (a triangle with a further triangle attached along each of its edges), the strip of four triangles, and the triangular prism minus one of its three matching edges (Section 7). We have not found Theorem 1.2 stated in the literature.

The proof has two steps, both relying on ideas of [5]. The first (Lemma 3.1) is the averaging argument of [5, Claim 4.4], stated as a general embedding result: if a graph $\Gamma$ on N vertices has at most $\lambda e\left(\Gamma \right)$ triangles, then $\overline{\Gamma }$ contains every graph without isolated vertices having at most $N/(624\lambda +56)$ edges. The second (Lemma 4.1) uses the triangle-elimination process of [5, Section 4]: if H is a 2-tree on h vertices, the process eliminates every triangle of an H-free graph $\Gamma$, so $\Gamma$ has at most $(h-3)e\left(\Gamma \right)$ triangles. In [5] these tools are used to find subdivisions of $K_4$; the point here is that for 2-trees they suffice on their own.

Consequences

Corollary 1.3.

Let G be a connected graph with $e\left(G\right)\le v\left(G\right)+2$.

(a)

If G contains neither $K_4$ nor $K_4^{*}$, then G is Ramsey size-linear.

(b)

If G contains $K_4$, then G is not Ramsey size-linear.

(c)

If G contains $K_4^{*}$, then G is Ramsey size-linear if and only if $K_4^{*}$ is.

Thus the threshold $v\left(G\right)+1$ of fact (i), which is sharp because of $K_4$, can be raised to $v\left(G\right)+2$ at the cost of a single unresolved graph, $K_4^{*}$.

For $k\ge 3$ let $W_k=K_1+C_k$ be the wheel with k spokes, and for $k\ge 4$ let $W_k^{-}$ denote $W_k$ with one spoke removed; see Figure 1.

Proposition 1.4.

Let $k\ge 4$.

(a)

$W_k$ is not Ramsey size-linear, while $W_k$ minus a rim edge is. Consequently, $W_k$ is minimally non-Ramsey size-linear if and only if $W_k^{-}$ is Ramsey size-linear.

(b)

$W_4^{-}\cong K_4^{*}$.

(c)

$m_2\left(W_k^{-}\right)=2$. Hence an affirmative answer to Question 1.1 would imply that every wheel $W_k$ with $k\ge 4$ is minimally non-Ramsey size-linear.

(d)

For $k\in \{4,5\}$ every proper subgraph of $W_k^{-}$ is Ramsey size-linear. Consequently each of the pairs $\{K_4^{*},W_4\}$ and $\{W_5^{-},W_5\}$ contains a minimally non-Ramsey size-linear graph. In particular, there are at least two minimally non-Ramsey size-linear graphs other than $K_4$: one of $K_4^{*},W_4$ on five vertices and one of $W_5^{-},W_5$ on six vertices.

Proposition 1.4 does not depend on Theorem 1.2: it follows from facts (i)–(iii) and [5]. Erdős et al. already observed that $K_4^{*}$ would be minimally non-Ramsey size-linear if it were not Ramsey size-linear [12]; the new point is the other member of each pair. Part (d) does not settle Wigderson’s problem, since we cannot say which member of each pair is minimally non-Ramsey size-linear. It does show that any resolution of the question of Erdős et al. for $K_4^{*}$, in either direction, produces an explicit minimally non-Ramsey size-linear graph other than $K_4$: namely $W_4$ if $K_4^{*}$ is Ramsey size-linear, and $K_4^{*}$ itself otherwise. The same holds for $W_5^{-}$ and $W_5$. Our computations (Section 7) show that, with the tools currently available, these are the only such pairs among the density-critical graphs on at most eight vertices.

Finally, in Section 6 we make explicit what the proof of [5, Theorem 4] actually produces, and in Section 7 we determine by an exhaustive computer search the minimal graphs on at most eight vertices that satisfy the hypothesis of Question 1.1 but whose Ramsey size-linearity does not follow from the results available. Apart from $K_4^{*}$ there are 3, 10 and 35 of them on six, seven and eight vertices. The three on six vertices are $W_5^{-}$, the triangular prism and $K_{3,3}$, and all of them reduce to one of eleven cores (Proposition 7.2).

Organization.Section 2 collects preliminaries. Lemma 3.1 is proved in Section 3 and Lemma 4.1 in Section 4. Section 5 contains the proofs of Theorem 1.2, Corollary 1.3 and Proposition 1.4, Section 6 the consequences of the proof of [5], Section 7 the census of small cases, and Section 8 concluding remarks and open questions.

2. Preliminaries

All graphs are finite and simple. For a graph $\Gamma$ we write $v\left(\Gamma \right)$ and $e\left(\Gamma \right)$ for the numbers of vertices and edges, $N_{\Gamma }\left(x\right)$ and $d_{\Gamma }\left(x\right)$ for the neighbourhood and the degree of a vertex x, $T\left(\Gamma \right)$ for the number of triangles, and $\overline{\Gamma }$ for the complement. We say that $\Gamma$containsG if $\Gamma$ has a (not necessarily induced) subgraph isomorphic to G, and that $\Gamma$ is G-free otherwise. If $G\subseteq G^{\prime }$ then $R(G,H)\le R(G^{\prime },H)$; hence every subgraph of a Ramsey size-linear graph is Ramsey size-linear.

Treewidth.

A 2-tree is a graph obtained from a triangle by repeatedly adding a new vertex adjacent to both ends of an existing edge. A 2-tree on $k\ge 3$ vertices has $2k-3$ edges. We use the following standard facts; see [4,9,10].

(T1)

A graph has treewidth at most two if and only if it has no $K_4$-minor.

(T2)

Every graph of treewidth at most two with at least three vertices is a spanning subgraph of a 2-tree.

(T3)

A graph has a $K_4$-minor if and only if it contains a subdivision of $K_4$.

Lemma 2.1

(Bradač, Gishboliner and Sudakov [5, Lemma 2.3]). Let H be a graph with m edges, let $0\le k\le v\left(H\right)$, and let A be a set of k vertices of largest degree in H. Let Γ be a graph on N vertices with average degree d, and let ${\sigma }^{\prime }:A\to V\left(\Gamma \right)$ be an embedding of $H\left[A\right]$ into $\overline{\Gamma }$. If

$$N\ge {\displaystyle \frac{4m}{k+1}}max\left\{\underset{a\in A}{max}{d}_{\Gamma }\left({\sigma }^{\prime }\left(a\right)\right),2d\right\}+2v\left(H\right),$$

then ${\sigma }^{\prime }$ extends to an embedding of H into $\overline{\Gamma }$.

(included for completeness)).Proof (Proof Let D denote the maximum in the displayed inequality and let $W=\{u:d_{\Gamma }\left(u\right)\le 2d\}$. Fewer than $N/2$ vertices have degree larger than $2d$, so $\left|W\right|>N/2$. Embed the vertices of $V\left(H\right)\setminus A$ one at a time into W. When x is to be embedded, each of the k vertices of A has degree at least $d_H\left(x\right)$, so $(k+1)d_H\left(x\right)\le 2m$. The vertices that x must avoid are the images of the vertices already embedded (fewer than $v\left(H\right)$) and the $\Gamma$-neighbours of the images of its embedded H-neighbours; each such image lies in ${\sigma }^{\prime }\left(A\right)$ or in W and so has $\Gamma$-degree at most D. Hence fewer than $2mD/(k+1)+v\left(H\right)\le N/2<\left|W\right|$ vertices of W are excluded, and x can be embedded.    □

Lemma 2.2

(Erdős, Faudree, Rousseau and Schelp [12, Lemma 2 and Corollary 3]; see also [5, Lemma 5.3]). If G is connected and $G=G_1\cup G_2$ with $|V\left(G_1\right)\cap V\left(G_2\right)|=1$, then $R(G,H)\le R(G_1,H)+R(G_2,H)$ for every graph H without isolated vertices. Consequently, a connected graph is Ramsey size-linear if and only if all its blocks are.

For disjoint unions one has $R(G_1\cup G_2,H)\le R(G_1,H)+R(G_2,H)+v\left(G_1\right)$, so a graph is Ramsey size-linear if and only if all its blocks are, and it suffices to study 2-connected graphs.

3. Graphs with Few Triangles

The following lemma isolates the averaging argument of [5, Claim 4.4] as a general statement.

Lemma 3.1.

Let $\lambda \ge 0$ and let Γ be a graph on N vertices with $T\left(\Gamma \right)\le \lambda e\left(\Gamma \right)$. Let H be a graph with $m\ge 1$ edges and no isolated vertices. If $N\ge (624\lambda +56)m$, then $\overline{\Gamma }$ contains H.

Proof. 

If $e\left(\Gamma \right)=0$ then $\overline{\Gamma }$ is complete and contains H, as $N\ge 2m\ge v\left(H\right)$. So let $e=e\left(\Gamma \right)>0$ and let $d=2e/N$ be the average degree. For a vertex x let

$$N^{\prime }\left(x\right)=\{y\in N_{\Gamma }\left(x\right):d_{\Gamma }\left(y\right)\le d_{\Gamma }\left(x\right)\},d^{\prime }\left(x\right)=\left|N^{\prime }\left(x\right)\right|,t^{\prime }\left(x\right)=e\left(\Gamma \left[N^{\prime }\left(x\right)\right]\right).$$

For every edge $xy$ we have $y\in N^{\prime }\left(x\right)$ or $x\in N^{\prime }\left(y\right)$, so ${\sum }_x{d}^{\prime }\left(x\right)\ge e$. Every edge of $\Gamma \left[N^{\prime }\left(x\right)\right]$ forms a triangle with x, so ${\sum }_x{t}^{\prime }\left(x\right)\le 3T\left(\Gamma \right)\le 3\lambda e$. Suppose first that $\lambda >0$ and set

$$\Phi \left(x\right)=d^{\prime }\left(x\right)-{\displaystyle \frac{d_{\Gamma }\left(x\right)+2d}{13}}-{\displaystyle \frac{t^{\prime }\left(x\right)}{6\lambda }}.$$

Since ${\sum }_x{d}_{\Gamma }\left(x\right)=Nd=2e$, we get ${\sum }_x\Phi \left(x\right)\ge e-\frac{6e}{13}-\frac{e}{2}=\frac{e}{26}>0$. Fix x with $\Phi \left(x\right)>0$. Then $d^{\prime }\left(x\right)>0$ and

$$\left(\mathrm{a}\right)d_{\Gamma }\left(x\right)<13d^{\prime }\left(x\right),\left(\mathrm{b}\right)2d<13d^{\prime }\left(x\right),\left(\mathrm{c}\right)2t^{\prime }\left(x\right)/d^{\prime }\left(x\right)<12\lambda .$$

By (c) the graph $\Gamma \left[N^{\prime }\left(x\right)\right]$ has average degree less than $12\lambda$, so by the Caro–Wei (or Turán) bound it contains an independent set I with $\left|I\right|\ge d^{\prime }\left(x\right)/(12\lambda +1)\ge 1$. If $\lambda =0$ then $\Gamma$ is triangle-free, every $N^{\prime }\left(x\right)$ is independent, and the same conclusion holds with $I=N^{\prime }\left(x\right)$, using $\Phi \left(x\right)=d^{\prime }\left(x\right)-(d_{\Gamma }\left(x\right)+2d)/13$ and ${\sum }_x\Phi \left(x\right)\ge e-\frac{6}{13}e=\frac{7}{13}e>0$.

Every $y\in I$ satisfies $d_{\Gamma }\left(y\right)\le d_{\Gamma }\left(x\right)<13d^{\prime }\left(x\right)\le 13(12\lambda +1)\left|I\right|$, and also $2d<13(12\lambda +1)\left|I\right|$. If $\left|I\right|\ge v\left(H\right)$, then H embeds into $\overline{\Gamma }\left[I\right]$, which is complete. Otherwise put $k=\left|I\right|$, let A be a set of k vertices of largest degree in H, and let ${\sigma }^{\prime }:A\to I$ be any bijection; it embeds $H\left[A\right]$ into $\overline{\Gamma }$ because I is independent in $\Gamma$. The maximum in Lemma 2.1 is at most $13(12\lambda +1)k$, so

$${\displaystyle \frac{4m}{k+1}}·13(12\lambda +1)k+2v\left(H\right)<52(12\lambda +1)m+4m=(624\lambda +56)m\le N,$$

using $v\left(H\right)\le 2m$. Lemma 2.1 extends ${\sigma }^{\prime }$ to an embedding of H into $\overline{\Gamma }$.    □

Remark 3.2.

Lemma 3.1 yields the following sufficient condition: if there is ${\lambda }_G$ such that every G-free graph Γ satisfies $T\left(\Gamma \right)\le {\lambda }_{G}e\left(\Gamma \right)$, then $R(G,H)\le (624{\lambda }_G+56)e\left(H\right)$ and G is Ramsey size-linear. For $G=K_3$ it gives $R(K_3,H)\le 56e\left(H\right)$, much weaker than the sharp bound $2e\left(H\right)+1$ [14,18]; no attempt has been made to optimise the constants.

4. Triangles in Graphs Without a Given 2-Tree

Lemma 4.1.

Let H be a 2-tree on $h\ge 3$ vertices. Then every H-free graph Γ satisfies $T\left(\Gamma \right)\le (h-3)e\left(\Gamma \right)$.

Proof. 

The elimination process below is the one of [5, Section 4], and the embedding step follows [5, Proposition 1.1]. Let $\mathcal{T}$ be the set of triangles of $\Gamma$. As long as some edge of $\Gamma$ lies in at least one and at most $h-3$ triangles of $\mathcal{T}$, choose such an edge and delete from $\mathcal{T}$ all triangles containing it. After this step the chosen edge lies in no triangle of $\mathcal{T}$, and $\mathcal{T}$ only shrinks, so each edge is chosen at most once. Hence at most $(h-3)e\left(\Gamma \right)$ triangles are deleted in total. Let ${\mathcal{T}}_0$ be the final set. By construction, every edge lying in some triangle of ${\mathcal{T}}_0$ lies in at least $h-2$ triangles of ${\mathcal{T}}_0$.

We claim ${\mathcal{T}}_0=\varnothing$, which gives $T\left(\Gamma \right)\le (h-3)e\left(\Gamma \right)$. Suppose not. Order $V\left(H\right)$ as $v_1,\dots ,v_h$ so that $v_1{v}_2{v}_3$ is a triangle and each $v_i$ with $i\ge 4$ is adjacent to exactly two earlier vertices $v_a,v_b$, which are adjacent. Map $v_1,v_2,v_3$ onto the vertices of a triangle of ${\mathcal{T}}_0$. Inductively, assume $v_1,\dots ,v_{i-1}$ are embedded injectively so that every edge of $H\left[\{v_1,\dots ,v_{i-1}\}\right]$ is mapped to an edge lying in a triangle of ${\mathcal{T}}_0$. The image of $v_a{v}_b$ then lies in at least $h-2$ triangles of ${\mathcal{T}}_0$, whose third vertices are distinct, while only $i-3\le h-3$ vertices other than the images of $v_a,v_b$ are already used. Choose an unused third vertex z and map $v_i$ to z; the two new edges belong to a triangle of ${\mathcal{T}}_0$, so the invariant is maintained. At the end we obtain a copy of H in $\Gamma$, a contradiction.    □

5. Proofs of the Main Results

Proof 

(Proof of Theorem 1.2). If $v\left(G\right)\le 2$ then $G\subseteq K_3$; otherwise, by (T1) and (T2), G is a spanning subgraph of a 2-tree $G^{*}$ on h vertices. In both cases $G\subseteq G^{*}$ for a 2-tree $G^{*}$ on h vertices. Let $\Gamma$ be a graph on $N\ge (624h-1816)m$ vertices that does not contain G. Then $\Gamma$ is $G^{*}$-free, so $T\left(\Gamma \right)\le (h-3)e\left(\Gamma \right)$ by Lemma 4.1, and Lemma 3.1 with $\lambda =h-3$ shows that $\overline{\Gamma }$ contains H, because $624(h-3)+56=624h-1816$.    □

Proof 

(Proof of Corollary 1.3). (b) follows from fact (ii), since $e\left(K_4\right)=2·4-2$. For (a) and (c), note first that if a connected graph G contains a connected subgraph S, then $e\left(G\right)-v\left(G\right)\ge e\left(S\right)-v\left(S\right)$: G arises from S by adding edges and by adding vertices each joined by at least one edge. If G has no $K_4$-minor, it is Ramsey size-linear by Theorem 1.2. Otherwise, by (T3), G contains a subdivision S of $K_4$, and $e\left(S\right)-v\left(S\right)=2\ge e\left(G\right)-v\left(G\right)$. Hence equality holds, which means that every edge of G outside S is a bridge: S is the only block of G with a cycle, and the other blocks are single edges. Now $S\ne K_4$ in case (a), and $S\neg \cong K_4^{*}$ unless G contains $K_4^{*}$; the only subdivisions of $K_4$ with fewer than six vertices are $K_4$ and $K_4^{*}$. So in case (a), S has at least six vertices and is Ramsey size-linear by [5]; single edges are Ramsey size-linear, and Lemma 2.2 shows that G is Ramsey size-linear. In case (c), the 2-connected graph $K_4^{*}$ lies in a block of G, hence in S; since S is 2-connected and $e\left(S\right)-v\left(S\right)=e\left(K_4^{*}\right)-v\left(K_4^{*}\right)$, the argument above (applied to $K_4^{*}\subseteq S$) shows $S=K_4^{*}$. Thus G is obtained from $K_4^{*}$ by attaching bridges, and it is Ramsey size-linear if $K_4^{*}$ is, by Lemma 2.2; the converse holds because $K_4^{*}\subseteq G$.    □

Proof 

(Proof of Proposition 1.4). Let c be the hub of $W_k$ and $u_1{u}_2\dots u_k{u}_1$ its rim.

(a) $W_k$ has $k+1$ vertices and $2k=2(k+1)-2$ edges, so it is not Ramsey size-linear by fact (ii). Deleting the rim edge $u_k{u}_1$ leaves the fan $K_1+P_k$, which is Ramsey size-linear by fact (iii). Every proper subgraph of $W_k$ is contained in $W_k-f$ for some edge f, and for $k\ge 4$ the automorphisms of $W_k$ act transitively on the spokes and on the rim edges. Hence all proper subgraphs of $W_k$ are Ramsey size-linear if and only if $W_k^{-}$ is.

(b) Deleting the spoke $c{u}_4$ from $W_4$ leaves the complete graph on $\{c,u_1,u_2,u_3\}$ minus the edge $u_1{u}_3$, together with the path $u_1{u}_4{u}_3$; this is $K_4$ with the edge $u_1{u}_3$ subdivided by $u_4$.

(c) $W_k^{-}$ has $k+1$ vertices and $2k-1=2(k+1)-3$ edges. Let S be a proper subset of its vertices with $\left|S\right|=j\ge 2$. If $c\notin S$, then S spans at most j edges of the rim cycle if $j=k$, and at most $j-1$ edges otherwise; in both cases at most $2j-3$ edges when $j\ge 3$, and at most 1 edge when $j=2$. If $c\in S$, then $S\setminus \left\{c\right\}$ is a proper subset of the rim, so it spans a linear forest with at most $j-2$ edges, and together with at most $j-1$ spokes S spans at most $2j-3$ edges. Thus $m_2\left(W_k^{-}\right)=2$, and the second statement follows from (a).

(d) For $k=4$: every proper subgraph of $K_4^{*}$ is contained in $K_4^{*}-f$ for an edge f, which is connected (as $K_4^{*}$ is 2-connected) and has $6=v+1$ edges, so it is Ramsey size-linear by fact (i). For $k=5$, let $W_5^{-}=W_5-c{u}_1$. If f is a rim edge, then deleting c from $W_5^{-}-f$ leaves a path, so $W_5^{-}-f$ is a subgraph of $K_1+P_5$ and is Ramsey size-linear by fact (iii). If f is a spoke, then in $W_5^{-}-f$ the hub is adjacent to exactly three vertices of the rim $C_5$, so $W_5^{-}-f$ is a subdivision of $K_4$ with six vertices (branch vertices: c and its three neighbours), which is Ramsey size-linear by [5]. Hence every proper subgraph of $W_k^{-}$ is Ramsey size-linear for $k\in \{4,5\}$.

Now fix $k\in \{4,5\}$. If $W_k^{-}$ is not Ramsey size-linear, then it is minimally non-Ramsey size-linear, since all its proper subgraphs are Ramsey size-linear. If $W_k^{-}$ is Ramsey size-linear, then $W_k$ is minimally non-Ramsey size-linear by (a). In either case the pair $\{W_k^{-},W_k\}$ contains a minimally non-Ramsey size-linear graph; the two pairs are disjoint and do not contain $K_4$, and $W_4^{-}\cong K_4^{*}$ by (b).    □

6. More from the Proof of Bradač, Gishboliner and Sudakov

The proof of [5, Theorem 4] produces more than its statement asserts. We make this explicit, since it settles many small cases (Section 7). Throughout this section F is a graph with $m\ge 1$ edges and no isolated vertices, $\Gamma$ is a graph on $N\ge Cm$ vertices whose complement contains no copy of F, and $C\gg C_1\gg C_0$ are the constants of [5, Section 4]. Let $G_{+}$ be the set of edges lying in a triangle that survives the triangle-elimination process of Section 4 (run with threshold $C_0$), so that

$$\mathrm{every}\mathrm{edge}\mathrm{of}{G}_{+}\mathrm{lies}\mathrm{in}\mathrm{at}\mathrm{least}{C}_0\mathrm{triangles}\mathrm{all}\mathrm{of}\mathrm{whose}\mathrm{edges}\mathrm{are}\mathrm{in}{G}_{+}.$$

(1)

Lemma 6.1.

Let $B\subseteq \Gamma$ with $E\left(B\right)\subseteq G_{+}$, and let K be obtained from B by repeatedly adding a new vertex joined to both ends of an existing edge. If $v\left(K\right)<C_0$, then Γ contains a copy of K extending the copy of B.

Proof. 

Add the vertices of $K\setminus B$ in the order of the construction. When a vertex is to be joined to the ends of an edge e, the copy of e lies in $G_{+}$, so by (1) it lies in at least $C_0>v\left(K\right)$ triangles with all edges in $G_{+}$; their apexes are distinct, so one of them is unused. The two new edges lie in $G_{+}$, so the invariant is maintained.    □

Write $B_1(t_1,t_2)$ for the graph with vertices v, $u_1$, $u_2$, w, $z_1,\dots ,z_{t_1}$, $y_1,\dots ,y_{t_2}$ and edges $v{u}_1$, $v{u}_2$, $vw$, $u_1{u}_2$, $u_1{z}_i,w{z}_i$$(i\le t_1)$ and $u_2{y}_j,w{y}_j$$(j\le t_2)$. For a multigraph Q on a set A of size s, write $B_2(s,Q)$ for the graph with vertices $v_1,v_2$, A and one vertex $x_f$ for each $f\in E\left(Q\right)$, and edges $v_{1}a,v_{2}a$$(a\in A)$ together with the two edges joining $x_f$ to the ends of f. Finally, let $c_1$, $c_2$, $c_3$ be $K_4$ on $\{x,y,z,w\}$, the graph $K_4^{*}$ obtained from it by subdividing $xy$ once, and the graph $H_3$ obtained from it by subdividing $xy$ twice. In $c_1$ the edge $xy$ is called dead, in $c_3$ the edge of the subdivision path incident to y is called dead, and all other edges are live; in $c_2$ all edges are live.

Proposition 6.2.

With Γ as above and any fixed $t_1,t_2,s$ and Q:

(a)

Γ contains a copy of $B_1(t_1,t_2)$ all of whose edges lie in $G_{+}$;

(b)

Γ contains a copy of $B_2(s,Q)$ all of whose edges lie in $G_{+}$;

(c)

Γ contains a copy of $c_1$, of $c_2$ or of $c_3$ in which all live edges lie in $G_{+}$.

Proof. 

Part (a) is what Case 1 of the proof of [5, Theorem 4] produces: a vertex v, an edge $u_1{u}_2$ of $G_{+}$ with $u_1,u_2\in N_{+}\left(v\right)$, and $w\in N_{+}\left(v\right)$ such that $u_1,w$ and $u_2,w$ each have more than $C_0$ common neighbours in $G_{+}$; choosing $t_1+t_2$ distinct such common neighbours gives the copy. Part (b) is Case 2: it produces $v_1,v_2$ and a set $A^{\prime }$ of common $G_{+}$-neighbours of $v_1$ and $v_2$ such that every pair in $A^{\prime }$ has more than $C_0$ common neighbours in $G_{+}$; there $|A^{\prime }|\ge 3$, and the same averaging gives $|A^{\prime }|\ge s$ once $C_1$ is large in terms of s and $C_0$. Choosing a distinct such common neighbour for each edge of Q gives the copy. Part (c) is the case analysis at the end of Case 3, whose possible outcomes are a copy of $K_4$ in which all edges except the one joining the two tips of a diamond lie in $G_{+}$, a copy of $K_4^{*}$ in $G_{+}$, or a copy of $H_3$ in which all edges except one end edge of the subdivision path lie in $G_{+}$. These arguments use only [5, Claims 4.1 and 4.4], which hold for every $\Gamma$ as above, and not the graph whose copy is being sought; hence they apply simultaneously.    □

Corollary 6.3.

Every subgraph of a graph obtained from some $B_1(t_1,t_2)$ or some $B_2(s,Q)$ by 2-tree steps is Ramsey size-linear. So is every graph which, for each $i\in \{1,2,3\}$, is a subgraph of a graph obtained from $c_i$ by 2-tree steps along live edges and along edges created by earlier steps.

Proof. 

Combine Proposition 6.2 with Lemma 6.1 (applied to the live edges), taking $C_0$ larger than the order of the graph sought, and use that subgraphs of Ramsey size-linear graphs are Ramsey size-linear.    □

Corollary 6.3 contains [5, Theorem 4], which is how that theorem is proved: a subdivision of $K_4$ on at least six vertices is a subdivision of $H_1$, $H_2$ or $H_3$, and a subdivided edge $xy$ is a subgraph of a fan with hub y glued along $xy$ by 2-tree steps.

7. The Remaining Small Cases

By Lemma 2.2 it suffices to consider 2-connected graphs. The next proposition describes the minimal graphs not covered by Theorem 1.2 and [5]. An S-ear of a subgraph S is a path of length at least one whose two (distinct) ends lie in S and whose inner vertices and edges do not belong to S.

Proposition 7.1.

Let G be a 2-connected graph with $m_2\left(G\right)\le 2$ which has a $K_4$-minor and is not a subdivision of $K_4$ with at least six vertices. Then G contains $K_4^{*}$, or G contains a subgraph $S\cup P$, where S is a subdivision of $K_4$ with at least six vertices and P is an S-ear. Every such $S\cup P$ is 2-connected, has a $K_4$-minor and satisfies $e(S\cup P)=v(S\cup P)+3$.

Proof. 

By (T3), G contains a subdivision S of $K_4$. We have $S\neg \cong K_4$ because $m_2\left(K_4\right)>2$. If $S\cong K_4^{*}$ we are done; otherwise $v\left(S\right)\ge 6$, and $S\ne G$ by assumption. If some edge of G outside S joins two vertices of S, it is an S-ear. Otherwise G has a vertex outside S; let C be a component of $G-V\left(S\right)$. Since G is 2-connected, C has neighbours in at least two vertices $x\ne y$ of S, and a path from x to y through C is an S-ear. The last sentence is clear, since adding an ear increases $e-v$ by one.    □

Consequently, if only Theorem 1.2 and [5] are used, every minimal unresolved graph other than $K_4^{*}$ has the form $S\cup P$ and has exactly $v+3$ edges; Turán-type results can create exceptions, such as $Q_3$ below. By the path-extension result of [2], $S\cup P$ is Ramsey size-linear whenever P has length at least $d_S(x,y)+3$, where $x,y$ are its ends, so only short ears matter.

Computation.

We enumerated all 2-connected graphs G with at most eight vertices and $m_2\left(G\right)\le 2$. For $v\left(G\right)\le 7$ we used the atlas of all graphs on at most seven vertices, as implemented in NetworkX. For $v\left(G\right)=8$ we used that such a graph has minimum degree 2 or 3 (as $e\left(G\right)\le 2v\left(G\right)-3$), so it is obtained from a graph on seven vertices with $m_2\le 2$ by adding a vertex of degree 2 or 3; isomorphism classes were separated by Weisfeiler–Leman hashes followed by exact isomorphism tests. An independent enumeration, adding a vertex with an arbitrary neighbourhood to every graph on seven vertices, gave the same number, 1224. Each graph was classified by the following sufficient conditions for Ramsey size-linearity, all taken from the literature except (C3): (C1) $e\left(G\right)\le v\left(G\right)+1$ [12, Theorem 4]; (C2) $G-v$ is a forest for some vertex v, that is, G is a subgraph of some $K_1+T$ [12, Theorem 3]; (C3) treewidth at most two (Theorem 1.2); (C4) G is a subdivision of $K_4$ with at least six vertices [5]; (C5) G is bipartite and all vertices of one colour class have degree at most two, whence its Turán number is $O\left(n^{3/2}\right)$ [1,13]; (C6) G is a subgraph of $K_{3,3}-e$ or of $Q_3-e$, whose Turán numbers are stated to be $O\left(n^{3/2}\right)$ in [12] (for (C5) and (C6) we use [12, Theorem 5]); (C7) G arises from a graph whose blocks satisfy one of the previous conditions by adding a path of length at least $d(x,y)+3$ between two of its vertices $x,y$ [2, Corollary 2]; (C8) G is a subgraph of a graph satisfying one of (C1), (C2) or (C4)–(C7); (C9) G is a subgraph of a 2-tree extension of some $B_1(t_1,t_2)$ or $B_2(s,Q)$ (Corollary 6.3); (C10) G lies in the extension class of each of $c_1,c_2,c_3$ (Corollary 6.3). Treewidth was tested by series–parallel reduction. A graph is unresolved if none of (C1)–(C8) applies, and minimal unresolved if moreover all its proper subgraphs are resolved. Conditions (C9) and (C10) were tested by a reduction: a graph lies in the extension class of a base B if and only if it can be reduced to a subgraph of B by repeatedly deleting a vertex of degree at most two and, when its degree is two, adding the edge between its two neighbours (an edge which must be live). Table 1 separates the contributions of (C3) and of (C9)–(C10).

On five vertices Theorem 1.2 adds nothing: Erdős et al. had already settled every graph on at most five vertices except $K_4^{*}$ [12, Section 3]. On six vertices it settles exactly three graphs not covered before, namely the 3-sun, the strip of four triangles and the prism minus a matching edge.

After all ten conditions, 49 minimal unresolved graphs remain: $K_4^{*}$; on six vertices $W_5^{-}$, the triangular prism and $K_{3,3}$; and 10 and 35 graphs on seven and eight vertices, listed in Appendix A. They are governed by few cores. Call a graph a core if it has minimum degree at least three, and reduce a graph by repeatedly deleting a vertex of degree at most two, adding the edge between its two neighbours when the degree is two; each step undoes a 2-tree step, possibly after deleting edges.

Proposition 7.2.

Each of the 49 minimal unresolved graphs on at most eight vertices reduces to exactly one of the following eleven cores (listed with the number of graphs reducing to it):

• six cores which arenotRamsey size-linear, having $2v-2$ edges: $K_4$ (5 graphs), the wheel $W_4$ (22 graphs), and four graphs on six vertices with ten edges (1 graph each);
• five cores with $m_2=2$ and minimum degree three: the triangular prism (10 graphs), $K_{3,3}$ (5), the cube $Q_3$, a graph on seven vertices with eleven edges, and a cubic graph on eight vertices (1 graph each).

In particular $K_4^{*}$ and $W_5^{-}$ are the smallest members of the first group: they are obtained from the non-Ramsey size-linear graphs $K_4$ and $W_4$ by subdividing one edge. The second group contains the graphs $K_{3,3}$ and $Q_3$ of [12, Question 2]. No graph of either group lies in one of the extension classes of Corollary 6.3, so settling any of them requires a new argument.

Density-critical graphs.

Call Gdensity-critical if $e\left(G\right)=2v\left(G\right)-2$ and every proper subset S of $V\left(G\right)$ with $\left|S\right|\ge 2$ spans at most $2\left|S\right|-3$ edges. By fact (ii) such a graph is not Ramsey size-linear, and every minimal non-Ramsey size-linear graph G with $m_2\left(G\right)>2$ is density-critical. A density-critical graph has minimum degree exactly three. There are 1, 1, 4, 15 and 109 of them on $4,\dots ,8$ vertices. For each, we determined the unresolved 2-connected proper subgraphs (every proper subgraph lies in some $G-f$, and every 2-connected subgraph of $G-f$ lies in a block of $G-f$). The argument of Proposition 1.4(d) applies to G exactly when all of them are minimal unresolved; this happens only for $K_4$ (with no unresolved subgraphs), for $W_4$ (with $K_4^{*}$) and for $W_5$ (with $W_5^{-}$).

8. Concluding Remarks

8.0.0.4. A programme.

By Proposition 7.2, settling Question 1.1 for all graphs on at most eight vertices amounts to settling the families attached to eleven cores. For the six cores that are not Ramsey size-linear, the question is whether subdividing edges of such a core, just enough to bring $m_2$ down to 2, always yields Ramsey size-linear graphs; $K_4^{*}$ and $W_5^{-}$ are the smallest instances. For the five cores with $m_2=2$ the question concerns the cores themselves, and for $K_{3,3}$ it requires $R(K_{3,3},K_n)=O\left(n^2\right)$. Testing each of the 49 graphs against the three configurations of Proposition 6.2(c) separately, with the dead edges declared live or not, shows how far the method is from each of them. For 45 graphs (all families except that of $K_4$) none of the three extension classes applies even when every edge is live, so a new structure is needed. For $K_4^{*}$ and three other graphs with core $K_4$ the obstruction is the configuration $c_3$, even with all its edges live. For one graph on eight vertices, $H_3$ with a triangle glued along each end edge of its subdivision path (graph6 GJyGcc), the only obstruction is the dead edge of $c_3$; in the notation of [5, Case 3], it is present in $\Gamma$ unless the set U of tips induces a triangle-free graph all of whose edges have codegree at most seven.

Short cycles in links.

For a vertex a let $L_a=G_{+}\left[N_{+}\left(a\right)\right]$ be its link in $G_{+}$; by (1) it has minimum degree at least $C_0$. If $L_a$ contains a cycle of length $k\in \{3,4,5\}$, then a together with this cycle spans a wheel $W_k=K_1+C_k$ (with $W_3=K_4$) all of whose edges lie in $G_{+}$, and Lemma 6.1 applies to it. Testing the 49 minimal unresolved graphs against the extension classes of $W_3,W_4,W_5$ gives the following.

• If some link contains a cycle of length at most five, whichever length it is, then $\Gamma$ contains GJyGcc. Equivalently, if $\Gamma$ does not contain GJyGcc, then every link has girth at least six; this can also be checked by hand (a $C_3$ in $L_a$ gives a $K_4$ in $G_{+}$; a $C_4$ gives two tips of a diamond in $G_{+}$ with a third common neighbour; a $C_5$ gives the required path between the tips).
• If some link contains a $C_4$ or a $C_5$, then $\Gamma$ contains $W_5^{-}$, 19 further graphs with core $W_4$, and GJyGcc.
• If some link contains a $C_3$ or a $C_4$, then $\Gamma$ contains $K_4^{*}$, GJyGcc and two further graphs with core $K_4$.

In the quotients of ${\tilde{A}}_2$-buildings of Remark 8.3 every link is the incidence graph of a projective plane and has girth six, so none of these hypotheses follows from (1) alone; a proof would have to use [5, Claim 4.1]. For $K_4^{*}$ the third item is little more than a reformulation, since every neighbourhood in a $K_4^{*}$-free graph is $C_4$-free; for $W_5^{-}$ and GJyGcc it is a genuine reduction.

Question 8.1.

Let Γ be as in Section 6. Must some link $L_a$ contain a cycle of length at most five? Must some link contain a cycle of length four or five?

A positive answer to the second part would show that $W_5^{-}$ is Ramsey size-linear and hence, by Proposition 1.4, that $W_5$ is an explicit minimally non-Ramsey size-linear graph.

8.0.0.6. The graph $W_5^{-}$.

By Proposition 1.4, deciding whether $W_5^{-}$ is Ramsey size-linear would, in either direction, produce an explicit minimally non-Ramsey size-linear graph other than $K_4$: $W_5$ if the answer is positive, and $W_5^{-}$ itself (together with a negative answer to Question 1.1) otherwise. The same is true for $K_4^{*}$, but $K_4^{*}$ has resisted several attempts [2,5,12], and $W_5^{-}$, whose rim is longer, may be more accessible. Natural intermediate steps, in analogy with [5], are the following.

Question 8.2.

Is $R(W_5^{-},F)=O\left(e\left(F\right)\right)$ for every bipartite graph F without isolated vertices? Is $R(W_5^{-},K_n)=O\left(n^2\right)$?

The general bounds of [5] give $R(W_5^{-},K_n)=O\left(n^3\right)$.

Remark 8.3

(Limits of the method; sketch). The only property of G used in the proof of Theorem 1.2 is that G-free graphs have $O\left(e\right)$ triangles. This fails for $K_4$ (consider complete tripartite graphs) and, less obviously, for $W_5^{-}$. Let Δ be the Bruhat–Tits building of ${\mathrm{PGL}}_3\left({\mathbb{F}}_q\left(\left(t\right)\right)\right)$. Its vertex links are point–line incidence graphs of the projective plane $\mathrm{PG}(2,q)$ [17], so every edge of its 1-skeleton lies in exactly $q+1$ triangles and every neighbourhood induces a bipartite graph of girth six. Suppose Δ contained $W_5^{-}$, with hub c, rim path $u_1{u}_2{u}_3{u}_4$ in the neighbourhood of c, and a further common neighbour $w\ne c$ of $u_1$ and $u_4$. By girth six, $u_1$ and $u_4$ are at distance three in the link of c, i.e., opposite. In the CAT(0) metric of Δ [6], where all triangles are equilateral with side one, the path $u_{1}c{u}_4$ is then a geodesic of length 2, and $u_{1}w{u}_4$ would be a second geodesic of the same length between the same points, contradicting the uniqueness of geodesics. Cocompact lattices in ${\mathrm{PGL}}_3\left({\mathbb{F}}_q\left(\left(t\right)\right)\right)$ exist (see e.g., [8]) and are residually finite, being finitely generated linear groups; hence Δ has finite quotients in which every ball of radius two is isomorphic to a ball of Δ. Their 1-skeleta are finite $W_5^{-}$-free graphs with $T/e=(q+1)/3$. Thus proving that $W_5^{-}$ is Ramsey size-linear requires more than triangle counting.

Question 8.4.

For which graphs G is there a constant ${\lambda }_G$ such that every G-free graph Γ satisfies $T\left(\Gamma \right)\le {\lambda }_{G}e\left(\Gamma \right)$? By Lemma 4.1 this holds when G has treewidth at most two. Does it fail for every graph with a $K_4$-minor?

Constants.

The constant $624h$ in Theorem 1.2 is far from optimal; for $G=K_3$ the truth is $2e\left(H\right)+1$. It would be interesting to determine the order of magnitude of the best constant for 2-trees on h vertices.