Ramsey Numbers of Sparse Graphs Versus Bipartite Graphs

1. Introduction

1.1. Background

For graphs G and H, the Ramsey number $R(G,H)$ is the least N such that every graph on N vertices contains G or has a complement containing H. A graph G is Ramsey size-linear if there is a constant $C_G$ with $R(G,H)\le C_{G}e\left(H\right)$ for every graph H without isolated vertices. This notion was introduced by Erdős, Faudree, Rousseau and Schelp [9]. They showed that a graph with $e\left(G\right)\ge 2v\left(G\right)-2$ is not Ramsey size-linear, and asked whether every graph in which each j-vertex subgraph spans at most $2j-3$ edges is Ramsey size-linear; this is Erdős Problem #566 [3]. The triangle is Ramsey size-linear by the theorems of Sidorenko [15] and Goddard and Kleitman [10], and several operations preserving Ramsey size-linearity are known [1,9]. Bradač, Gishboliner and Sudakov [5] proved, among other results, that every subdivision of $K_4$ with at least six vertices is Ramsey size-linear. Wigderson [18] showed that there are infinitely many minimally non-Ramsey size-linear graphs. It is shown in [2] that every graph without a $K_4$-minor is Ramsey size-linear, together with a computer reduction of the open cases on at most eight vertices to 49 minimal graphs. For a survey of graph Ramsey theory we refer to [7].

Since $e\left(K_{n,n}\right)=n^2$, a positive answer to Problem #566 would give $R(G,K_{n,n})=O\left(n^2\right)$ for all such G. Bradač, Gishboliner and Sudakov [5] asked whether $R(H,K_{n,n})=O\left(n^2\right)$ holds for every 2-degenerate graph H. This question is a bipartite, off-diagonal analogue of the Burr–Erdős conjecture [6], proved by Lee [11], that degenerate graphs have Ramsey numbers linear in their number of vertices. They proved $R(K_4^{*},F)=O\left(e\left(F\right)\right)$ for every bipartite F, where $K_4^{*}$ is $K_4$ with one edge subdivided, and $R(H,K_{n,n})=O\left(n^3\right)$ whenever $e\left(H\right)-v\left(H\right)\le 4$ [5].

1.2. Results

Let $W_k^{-}$ denote the wheel $W_k$ minus a spoke: a hub h adjacent to the vertices of a path $u_1{u}_2\dots u_{k-1}$, and a vertex w adjacent to $u_1$ and $u_{k-1}$ (Figure 1). It is 2-degenerate, and $W_4^{-}=K_4^{*}$.

Theorem 1. 

For all integers $k\ge 4$ and $n\ge 1$, $R(W_k^{-},K_{n,n})\le (8k+16)n^2$. 

For $k=4$ this recovers the case $F=K_{n,n}$ of the theorem of [5] on $K_4^{*}$. The graph $W_5^{-}$ is one of the 49 minimal open graphs of [2], and if it is Ramsey size-linear then $W_5$ is an explicit minimally non-Ramsey size-linear graph (see [2]). For general bipartite graphs we lose only a factor $\sqrt{loge\left(F\right)}$.

Theorem 2. 

There is an absolute constant C such that $R(W_5^{-},F)\le Ce\left(F\right)\sqrt{log\left(e\right(F)+1)}$ for every bipartite graph F without isolated vertices. 

The proof (Section 6) follows the three cases of [5] for $K_4^{*}$. The new ingredient is that in a $W_5^{-}$-free graph the edges lying in at least four triangles form a $C_4$-free graph inside every neighbourhood (Lemma 5), which recovers the triangle bound used in [5]. Combined with the bound of Sudakov [17] and Li and Zang [12] on $R(C_5,K_n)$, the same lemma gives $R(W_5^{-},K_n)=O(n^{5/2}/{log}^{3/2}n)$ (Remark 3).

Our next result concerns the graph J shown in Figure 1. It is obtained from $K_4$ by subdividing one edge twice and gluing a triangle along each end edge of the subdivision path, and it is another of the 49 minimal graphs of [2].

Theorem 3. 

For every integer $n\ge 1$, $R(J,K_{n,n})\le 20n^2$. 

Both proofs work inside a core of the host graph in which every vertex has degree at least $Kn$ (Lemma 2). In such a core, pairs of vertices with few common neighbours are rare (Lemma 1), which makes it possible to embed graphs greedily and to find configurations by double counting. Section 4 turns this into a general tool: we define construction sequences and show that $R(G,K_{n,n})\le 32v{\left(G\right)}^2{n}^2$ for every graph G admitting one (Theorem 5). As an application we test the method on the minimal open graphs of [2].

Theorem 4. 

Of the 49 minimal open graphs of [2], exactly 38 admit a construction sequence, and hence satisfy $R(G,K_{n,n})=O\left(n^2\right)$. They include all 27 graphs whose core is $K_4$ or $W_4$. The remaining eleven graphs are listed in Section 5. 

All 38 graphs are 2-degenerate, so Theorems 1–4 confirm the question of [5] for them. Every graph with a construction sequence has a vertex of degree at most two (Remark 1), so the method cannot reach graphs of minimum degree three, such as the triangular prism and $K_{3,3}$. Apart from this use of the list of minimal graphs in Section 5, the present paper does not depend on [2].

1.3. Methods

All results are proved by hand. The proofs combine the absence of n-holes with minimum-degree cores and double counting (Section 2, Section 3 and Section 4), and with the lemmas of [5] listed in Section 6. The only computer-assisted statement is Theorem 4. It rests on an exhaustive search for construction sequences, whose output is a list of certificates (Appendix A) that can be checked by hand and that was checked again by an independent program. All code and data are provided as Supplementary File S1. The large language model Claude (Anthropic) was used in the preparation of this work: for literature search, for developing and checking the proofs, for writing and running the programs of Section 5 and the random tests of Lemmas 5 and 6, and for drafting the text. The author verified every proof and every computational claim, and takes full responsibility for the content of the manuscript.

1.4. Organization

Section 2 collects properties of graphs without n-holes. Section 3 proves Theorems 1 and 3. Section 4 introduces construction sequences, and Section 5 applies them to the minimal open graphs. Section 6 proves Theorem 2 and the bound for $R(W_5^{-},K_n)$, and Section 7 discusses open problems.

Notation

All graphs are finite and simple. Throughout, $\mathsf{\Gamma }$ denotes the host graph and G, F the graphs to be found in it or in its complement. For vertices $x\ne y$ of a graph $\mathsf{\Gamma }$, $codeg(x,y)=\left|N\right(x)\cap N(y\left)\right|$. An n-hole in $\mathsf{\Gamma }$ is a pair of disjoint sets $A,B$ with $\left|A\right|=\left|B\right|=n$ and no edge between them. The complement $\overline{\mathsf{\Gamma }}$ contains $K_{n,n}$ if and only if $\mathsf{\Gamma }$ has an n-hole, so $R(G,K_{n,n})\le N$ means that every graph on N vertices without an n-hole contains G.

2. Graphs Without n-Holes

Lemma 1. 

Let Γ be a graph without an n-hole.

(a)

If $\left|S\right|\ge n$, then fewer than n vertices outside S have no neighbour in S. 

(b)

Every induced subgraph of Γ has independence number less than $2n$. 

(c)

Let $r\ge 1$. If $d\left(a\right)\ge max\{2rn,4r\}$, then fewer than $4n$ non-neighbours $y\ne a$ of a satisfy $codeg(a,y)\le r$. 

(d)

Let $K\ge 3$. If $d\left(v\right)\ge Kn$, then fewer than n neighbours x of v satisfy $codeg(v,x)<K-2$. 

Proof. (a) Take $S^{\prime }\subseteq S$ with $|S^{\prime }|=n$; n vertices outside S without neighbours in S would form an n-hole with $S^{\prime }$. (b) An independent set of size $2n$ splits into an n-hole.

(c) Let $X=N\left(a\right)$ and let Y be the set of non-neighbours $y\ne a$ with $codeg(a,y)\le r$; each $y\in Y$ has at most r neighbours in X. Choose $X^{\prime }\subseteq X$ uniformly at random with $|X^{\prime }|=t:=\lceil |X|/\left(2r\right)\rceil \ge n$. For $y\in Y$, the probability that $X^{\prime }$ contains a neighbour of y is at most $rt/\left|X\right|\le 1/2+r/\left|X\right|\le 3/4$. Hence the expected number of vertices of Y with no neighbour in $X^{\prime }$ is at least $\left|Y\right|/4$, and if $\left|Y\right|\ge 4n$, then for some choice of $X^{\prime }$ at least n vertices of Y have no neighbour in $X^{\prime }$; any n vertices of $X^{\prime }$ together with n such vertices of Y form an n-hole (the two sets are disjoint because $X^{\prime }\subseteq N\left(a\right)$ and $Y\cap N\left(a\right)=\varnothing$).

(d) Let $Q=\mathsf{\Gamma }\left[N\right(v\left)\right]$, so that ${deg}_Q\left(x\right)=codeg(v,x)$ for $x\in N\left(v\right)$, and Q has at least $Kn$ vertices and no n-hole. If Q had a set S of n vertices of Q-degree less than $K-2$, then by (a) inside Q at least $\left|Q\right|-2n+1\ge (K-2)n+1$ vertices of $Q\setminus S$ would have a neighbour in S. This gives ${\sum }_{s\in S}{deg}_Q\left(s\right)>(K-2)n$, a contradiction.    □

Lemma 2. 

Let K be a positive integer. Every graph on $N\ge 2K{n}^2$ vertices without an n-hole has a nonempty induced subgraph of minimum degree at least $Kn$. 

Proof. 

Otherwise the graph is $(Kn-1)$-degenerate, hence $Kn$-colourable, and has an independent set of size at least $N/\left(Kn\right)\ge 2n$, contradicting Lemma 1(b).    □

Lemma 3. 

Let Q be a graph with $\alpha \left(Q\right)<2n$. Let τ be a tree on $s\ge 2$ vertices, let $u\ne v$ be vertices of τ with v a leaf, and let $K^{\prime }\ge s$ be an integer. Then at least $\left|Q\right|-2n{K}^{\prime }$ vertices x of Q satisfy the following: the set of images $\psi \left(v\right)$, over all embeddings ψ of τ into Q with $\psi \left(u\right)=x$, has at least $K^{\prime }-s+2$ elements. In particular, for paths: at least $\left|Q\right|-2n{K}^{\prime }$ vertices of Q are endpoints of paths with ℓ edges ending at $K^{\prime }-\ell +1$ or more distinct vertices, whenever $K^{\prime }\ge \ell +1$. 

Proof. 

Repeatedly delete a vertex of degree less than $K^{\prime }$ in the current graph, and let D be the set of deleted vertices. The graph $Q\left[D\right]$ is $(K^{\prime }-1)$-degenerate, hence $K^{\prime }$-colourable, so $\left|D\right|/K^{\prime }\le \alpha \left(Q\right)<2n$. The remaining graph $Q^{*}$ has minimum degree at least $K^{\prime }$. Order the vertices of $\tau$ as $u=w_1,\dots ,w_s=v$ so that the neighbour of each $w_j$ ($j\ge 2$) on the path to u comes earlier; this is possible because v is a leaf different from u. For $x\in V\left(Q^{*}\right)$ embed $w_1,w_2,\dots$ greedily into $Q^{*}$ with $\psi \left(w_1\right)=x$, choosing $\psi \left(w_j\right)$ among the neighbours of the image of the parent of $w_j$ that are not yet used; at most $j-2$ of them are used, so there are at least $K^{\prime }-s+2\ge 2$ choices at every step. At the last step the choices give at least $K^{\prime }-s+2$ distinct images of v.    □

3. Proofs of Theorems 1 and 3

Proof 

(Proof of Theorem 1). Let $k\ge 4$, put $K=4k+8$, and let $\mathsf{\Gamma }$ be a graph on $N\ge 2K{n}^2=(8k+16)n^2$ vertices without an n-hole. By Lemma 2 there is a nonempty induced subgraph ${\mathsf{\Gamma }}^{\prime }$ of minimum degree at least $Kn$; it has no n-hole either. Suppose for a contradiction that ${\mathsf{\Gamma }}^{\prime }$ does not contain $W_k^{-}$. From now on, neighbourhoods and codegrees are taken in ${\mathsf{\Gamma }}^{\prime }$, and $N^{\prime }=\left|V\left({\mathsf{\Gamma }}^{\prime }\right)\right|$.

Label $W_k^{-}$ by a hub adjacent to $u_1,\dots ,u_{k-1}$, a rim path $u_1{u}_2\dots u_{k-1}$, and a vertex w adjacent to $u_1$ and $u_{k-1}$ only. Since ${\mathsf{\Gamma }}^{\prime }$ has no $W_k^{-}$, for every vertex h and every path $u_1\dots u_{k-1}$ in ${\mathsf{\Gamma }}^{\prime }\left[N\left(h\right)\right]$ we have $N\left(u_1\right)\cap N\left(u_{k-1}\right)\subseteq \{h,u_2,\dots ,u_{k-2}\}$, and so

$$codeg(u_1,u_{k-1})\le k-2.$$

(1)

Call a pair $\{x,y\}$poor if $codeg(x,y)\le k-2$. A vertex x lies in fewer than $5n$ poor pairs. Indeed, fewer than n of them have $y\in N\left(x\right)$, by Lemma 1(d), since $k-2<K-2$. Fewer than $4n$ of them have $y\notin N\left(x\right)$, by Lemma 1(c) with $r=k-2$, which applies since $d\left(x\right)\ge Kn\ge max\left\{2\right(k-2)n,4(k-2\left)\right\}$. Hence there are fewer than ${\displaystyle \frac{5}{2}}n{N}^{\prime }$ poor pairs.

For a vertex h, let ${\pi }_h$ be the number of unordered pairs of vertices of $N\left(h\right)$ joined by a path with $k-2$ edges in ${\mathsf{\Gamma }}^{\prime }\left[N\left(h\right)\right]$. By (1) every such pair is poor, and it is counted for at most $k-2$ vertices h, all of which are common neighbours of the pair. Therefore

$$\sum _h{\pi }_h<\frac{5}{2}(k-2)n{N}^{\prime }.$$

On the other hand, $Q={\mathsf{\Gamma }}^{\prime }\left[N\left(h\right)\right]$ has at least $Kn$ vertices and $\alpha \left(Q\right)<2n$. Apply Lemma 3 to paths with $\ell =k-2$ edges and $K^{\prime }=k+2$: at least $\left|Q\right|-2n(k+2)\ge \left|Q\right|/2$ vertices of Q have at least $K^{\prime }-\ell +1=5$ partners. Hence ${\pi }_h\ge {\displaystyle \frac{5}{4}}\left|Q\right|\ge {\displaystyle \frac{5}{4}}Kn=5(k+2)n$, and ${\sum }_h{\pi }_h\ge 5(k+2)n{N}^{\prime }$. This contradicts the previous bound.    □

Proof 

(Proof of Theorem 3). Label J as follows: a diamond on $\{a,b,p,q\}$ with middle edge $ab$ and tips $p,q$; a path $pxyq$; a vertex s adjacent to p and x; and a vertex $s^{\prime }$ adjacent to y and q.

Let $\mathsf{\Gamma }$ be a graph on $N\ge 20n^2$ vertices without an n-hole, and let ${\mathsf{\Gamma }}^{\prime }$ be a nonempty induced subgraph of minimum degree at least $10n$ (Lemma 2 with $K=10$); we work in ${\mathsf{\Gamma }}^{\prime }$. For a vertex v let ${\mathrm{Low}}_v=\{z\in N\left(v\right):codeg(v,z)<8\}$; by Lemma 1(d), $|{\mathrm{Low}}_v|<n$.

The diamond. Take any vertex a. Since $\left|N\right(a\left)\right|\ge 10n>n$, there is $b\in N\left(a\right)\setminus {\mathrm{Low}}_a$, and b has two distinct neighbours $p,q$ in $N\left(a\right)$. Then $a,b,p,q$ span a diamond with middle edge $ab$ and tips $p,q$.

The path. Let $X=N\left(p\right)\setminus ({\mathrm{Low}}_p\cup \{a,b,q\})$ and $Y=N\left(q\right)\setminus ({\mathrm{Low}}_q\cup \{a,b,p\})$; both have at least $10n-n-3\ge 3n$ elements. If $|X\cap Y|\ge 2n$, then $X\cap Y$ contains an edge $xy$ by Lemma 1(b). Otherwise $X\setminus Y$ and $Y\setminus X$ both have at least n elements, so there is an edge $xy$ with $x\in X\setminus Y$ and $y\in Y\setminus X$. In both cases $x\in X$, $y\in Y$, $x\ne y$ and $xy$ is an edge.

The ears. Since $x\notin {\mathrm{Low}}_p$, we have $codeg(p,x)\ge 8$, so there is $s\in N\left(p\right)\cap N\left(x\right)\setminus \{a,b,q,y\}$. Since $y\notin {\mathrm{Low}}_q$, there is $s^{\prime }\in N\left(q\right)\cap N\left(y\right)\setminus \{a,b,p,x,s\}$.

The eight vertices are distinct, because $x\in N\left(p\right)\setminus \{a,b,q\}$, $y\in N\left(q\right)\setminus \{a,b,p\}$, $x\ne y$, and $s,s^{\prime }$ were chosen outside the other vertices. All twelve edges of J are present.    □

4. Construction Sequences

Definition 1. 

Let G be a graph. Aconstruction sequencefor G is a sequence of steps whose vertex sets partition $V\left(G\right)$. During the sequence we keep the set E of vertices placed so far and a set $\mathcal{P}$ ofmarkedpairs of placed vertices, both initially empty. The allowed steps are: 

Tree step $\left\{T\right\}$.

$T\subseteq V\left(G\right)\setminus E$ is nonempty, $G\left[T\right]$ is a tree, and every vertex of T has at most one neighbour in E. All pairs $\{x,e\}$ with $x\in T$, $e\in E$ become marked; then T is added to E. 

Pair step $g:f_1{f}_2$.

$g\notin E$, the neighbours of g in E are exactly $f_1,f_2$, and $\{f_1,f_2\}$ is marked; g is added to E. 

Wheel step $\mathrm{W}(h;T;u,v)$,

allowed only as the first step. Here $h\notin T$, $\left|T\right|\ge 2$, $G\left[T\right]$ is a forest, $u\ne v$ lie in T, and v has at most one neighbour in T. The pair $\{u,v\}$ and all pairs $\{h,x\}$ with $x\in T$ become marked, and $E=\left\{h\right\}\cup T$. 

A tree step with $\left|T\right|=1$ adds a vertex with at most one placed neighbour. The marked pairs record which pairs of placed vertices are guaranteed to have many common neighbours in the host graph.

Remark 1. 

The last step of a construction sequence places a vertex of degree at most two in G: the new vertex of a pair step, a leaf of $G\left[T\right]$ (or the single vertex) in a tree step, or the vertex v in a wheel step that places all of G. Hence graphs of minimum degree at least three have no construction sequence. 

Theorem 5. 

If a graph G on $t\ge 2$ vertices admits a construction sequence, then $R(G,K_{n,n})\le 32t^2{n}^2$ for every $n\ge 1$. 

The only step that is not a greedy choice is the wheel step, which is provided by the following double counting lemma. In it, a pair $\{x,y\}$ of distinct vertices is good if $codeg(x,y)>\kappa$ and bad otherwise.

Lemma 4. 

Let $\kappa \ge 1$, $s\ge 2$ and $K\ge 16\kappa +8s$. Let ${\mathsf{\Gamma }}^{\prime }$ be a graph without an n-hole and of minimum degree at least $Kn$, let τ be a tree on s vertices and let $u\ne v$ be vertices of τ with v a leaf. For a vertex h let ${\mathrm{Low}}_h=\{z\in N\left(h\right):codeg(h,z)\le \kappa \}$. Then there are a vertex h and an embedding ψ of τ into ${\mathsf{\Gamma }}^{\prime }[N\left(h\right)\setminus {\mathrm{Low}}_h]$ such that $\left\{\psi \right(u),\psi (v\left)\right\}$ is good. 

Proof. 

Every vertex x lies in fewer than $5n$ bad pairs: fewer than n with the other vertex in $N\left(x\right)$ by Lemma 1(d) (as $\kappa <K-2$), and fewer than $4n$ with the other vertex outside $N\left(x\right)$ by Lemma 1(c) with $r=\kappa$. So there are fewer than ${\displaystyle \frac{5}{2}}n{N}^{\prime }$ bad pairs, where $N^{\prime }=\left|V\left({\mathsf{\Gamma }}^{\prime }\right)\right|$; in particular $|{\mathrm{Low}}_h|<n$ for every h.

Suppose the lemma fails. For each h let $Q_h={\mathsf{\Gamma }}^{\prime }[N\left(h\right)\setminus {\mathrm{Low}}_h]$ and let ${\pi }_h$ be the number of unordered pairs $\left\{\psi \right(u),\psi (v\left)\right\}$ over all embeddings $\psi$ of $\tau$ into $Q_h$. Each such pair is bad and is counted only for vertices h in the common neighbourhood of the pair, so ${\sum }_h{\pi }_h<{\displaystyle \frac{5}{2}}\kappa n{N}^{\prime }$. On the other hand $|Q_h|\ge (K-1)n$ and $\alpha \left(Q_h\right)<2n$, so Lemma 3 with $K^{\prime }=2\kappa +s$ gives

$${\pi }_h\ge \frac{1}{2}\left(|Q_h|-2n{K}^{\prime }\right)(K^{\prime }-s+2)\ge \frac{1}{2}(K-1-4\kappa -2s)n(2\kappa +2)\ge \frac{5}{2}\kappa n·4,$$

using $K-1-4\kappa -2s\ge 10$. Summing over h gives a contradiction.    □

Proof 

(Proof of Theorem 5). Fix a construction sequence for G and put $\kappa =t$ and $K=16t^2$; note $K\ge 16\kappa +8t$ and $K\ge t^2+7t$. Let $\mathsf{\Gamma }$ be a graph on $N\ge 2K{n}^2=32t^2{n}^2$ vertices without an n-hole, and let ${\mathsf{\Gamma }}^{\prime }$ be an induced subgraph of minimum degree at least $Kn$ (Lemma 2). We work in ${\mathsf{\Gamma }}^{\prime }$ and use good and bad pairs as in Lemma 4. For a vertex z let $\mathrm{Bad}\left(z\right)$ be the set of vertices forming a bad pair with z; as shown there, $\left|\mathrm{Bad}\right(z\left)\right|<5n$.

We follow the sequence and build an injective map $\phi$ from the placed vertices into $V\left({\mathsf{\Gamma }}^{\prime }\right)$ that maps every edge of G between placed vertices to an edge and every marked pair to a good pair.

Wheel step. Extend the forest $G\left[T\right]$ to a tree $\tau$ on T in which v is still a leaf, by joining its components through vertices other than v (if v is isolated in $G\left[T\right]$, join it by a single edge). Lemma 4 gives $h^{\prime }$ and an embedding $\psi$ of $\tau$ into $N\left(h^{\prime }\right)\setminus {\mathrm{Low}}_{h^{\prime }}$ with $\left\{\psi \right(u),\psi (v\left)\right\}$ good. Put $\phi \left(h\right)=h^{\prime }$ and $\phi =\psi$ on T. The edges of G inside $\left\{h\right\}\cup T$ go from h to T or lie in $G\left[T\right]\subseteq \tau$, so they are mapped to edges. The marked pairs $\{h,x\}$ are good because $\psi \left(x\right)\notin {\mathrm{Low}}_{h^{\prime }}$.

Pair step $g:f_1{f}_2$. The pair $\{\phi \left(f_1\right),\phi \left(f_2\right)\}$ has more than $\kappa =t$ common neighbours, at most $t-2$ of which are already used. Map g to an unused one.

Tree step $\left\{T\right\}$. For $x\in T$ let $A_x$ be the neighbourhood of the image of the placed neighbour of x, or $V\left({\mathsf{\Gamma }}^{\prime }\right)$ if there is none, with the used vertices and the sets $\mathrm{Bad}\left(\phi \right(e\left)\right)$, $e\in E$, removed. Then $|A_x|\ge Kn-t-5tn\ge t(t+1)n$, so we can choose pairwise disjoint sets $A_x^{\prime }\subseteq A_x$ of size $(t+1)n$. Root the tree $G\left[T\right]$ at some vertex. Going from the leaves to the root, let $S_x$ be the set of vertices of $A_x^{\prime }$ that have a neighbour in $S_c$ for every child c of x. By induction $|S_c|>n$ for each child, so by Lemma 1(a) (note $S_c\cap A_x^{\prime }=\varnothing$), fewer than n vertices of $A_x^{\prime }$ are lost for each child, and $|S_x|>(t+1)n-tn=n$. Now map the root into its set S, and then, going down, map each child c of an already mapped vertex x to a neighbour of $\phi \left(x\right)$ in $S_c$. The images are distinct because the sets $A_x^{\prime }$ are disjoint, the edges of $G\left[T\right]$ and the edges to placed neighbours are mapped to edges, and every new vertex forms good pairs with all previously placed vertices because $\mathrm{Bad}\left(\phi \right(e\left)\right)$ was removed from $A_x$.

At the end all vertices and edges of G are mapped, so ${\mathsf{\Gamma }}^{\prime }$, and hence $\mathsf{\Gamma }$, contains G.    □

Theorem 1 is the case of the sequence $\mathrm{W}(h;\{u_1,\dots ,u_{k-1}\};u_1,u_{k-1}),w:u_1{u}_{k-1}$, and Theorem 3 the case of the sequence $\left\{a\right\},\left\{b\right\},p:ab,q:ab,\{x,y\},s:px,s^{\prime }:qy$; the direct proofs above give better constants.

5. The Minimal Open Graphs

For each of the 49 minimal open graphs of [2] we searched exhaustively for a construction sequence. The search is a depth-first search over the pairs $(E,\mathcal{P})$ with memoisation, preceded by every possible wheel step. Every sequence found was checked by an independent program, and the sequences are listed in Appendix A, so each case can be verified by hand. The outcome, grouped by core as in [2], is:

core	graphs	with a sequence	without
$K_4$	5	5	0
$W_4$	22	22	0
four graphs with 6 vertices and 10 edges	4	1	3
triangular prism	10	7	3
$K_{3,3}$	5	3	2
$Q_3$	1	0	1
the graph FJyKg (7 vertices, 11 edges)	1	0	1
the cubic graph GMs‵KK (8 vertices)	1	0	1
total	49	38	11

The eleven graphs without a construction sequence, in graph6 format, are EtTg (the prism), ElUg ($K_{3,3}$), FMohg, FlBHo, FhUk_, GhdWbC, GhUgMC, GhdYCS, GJyGcK, GMs‵KK and GhELQg ($Q_3$). Seven of them are 2-degenerate; the prism, $K_{3,3}$, $Q_3$ and the cubic graph GMs‵KK have minimum degree three.

The same search over all 925 graphs on at most eight vertices left open in [2] finds construction sequences for 254 of them (1, 4, 25 and 224 graphs on 5, 6, 7 and 8 vertices). Each of the remaining 671 graphs contains one of 68 minimal graphs without a construction sequence; 541 of the 671 are 2-degenerate, so they remain open instances of the question of [5].

6. $W_5^{-}$ Versus Arbitrary Bipartite Graphs

Bradač, Gishboliner and Sudakov proved $R(K_4^{*},F)=O\left(e\left(F\right)\right)$ for every bipartite F [5]. Their proof has three cases. The third uses that every neighbourhood in a $K_4^{*}$-free graph is $C_4$-free, which bounds the number of triangles and gives a logarithmic gain in the independence number. In a $W_5^{-}$-free graph neighbourhoods need not be $C_4$-free, but the following lemma recovers the same triangle bound.

Lemma 5. 

Let Γ be a graph without $W_5^{-}$, and call an edgestrongif its endpoints have at least four common neighbours.

(a)

For every vertex h and every path $u_1{u}_2{u}_3{u}_4$ in $\mathsf{\Gamma }\left[N\right(h\left)\right]$, $codeg(u_1,u_4)\le 3$. 

(b)

For every vertex h, the strong edges of $\mathsf{\Gamma }\left[N\right(h\left)\right]$ form a $C_4$-free graph. 

(c)

If $U\subseteq V\left(\mathsf{\Gamma }\right)$ and $\mathsf{\Delta }\left(\mathsf{\Gamma }\right[U\left]\right)\le D$, then $\mathsf{\Gamma }\left[U\right]$ has at most $3e\left(\mathsf{\Gamma }\left[U\right]\right)+\left|U\right|(D^{3/2}/6+D/12)$ triangles. 

Proof. (a) is (1) for $k=5$. (b) If $abcd$ were a 4-cycle of strong edges in $\mathsf{\Gamma }\left[N\right(h\left)\right]$, then $bcda$ would be a path in $\mathsf{\Gamma }\left[N\right(h\left)\right]$, and (a) would give $codeg(a,b)\le 3$, contradicting that $ab$ is strong. (c) An edge that is not strong lies in at most three triangles, so at most $3e\left(\mathsf{\Gamma }\right[U\left]\right)$ triangles of $\mathsf{\Gamma }\left[U\right]$ contain such an edge. In a triangle $abc$ of $\mathsf{\Gamma }\left[U\right]$ with three strong edges, $bc$ is a strong edge inside $N\left(a\right)\cap U$. By (b) and Reiman’s bound $\mathrm{ex}(n,C_4)\le {\displaystyle \frac{n}{4}}(1+\sqrt{4n-3})\le {\displaystyle \frac{1}{2}}{n}^{3/2}+{\displaystyle \frac{1}{4}}n$ [13], there are at most ${\displaystyle \frac{1}{3}}{\sum }_{a\in U}\mathrm{ex}\left(\right|N\left(a\right)\cap U|,C_4)\le |U|(D^{3/2}/6+D/12)$ such triangles.    □

Lemma 6. 

Every graph Q with M vertices has at least $e\left(Q\right)-2M$ unordered pairs of vertices joined by a path with exactly three edges. 

Proof. 

Order the vertices as $v_1,\dots ,v_M$ so that $v_i$ has minimum degree $f_i$ in $Q_i=Q\left[\{v_i,\dots ,v_M\}\right]$; then ${\sum }_i{f}_i=e\left(Q\right)$. If $f_i\ge 3$, choose a neighbour a of $v_i$ in $Q_i$ and a neighbour $b\ne v_i$ of a in $Q_i$; each of the at least $f_i-2$ neighbours $c\notin \{a,v_i\}$ of b in $Q_i$ is the end of the path $v_{i}abc$. So $v_i$ is joined in this way to at least $f_i-2$ vertices, all of which come after $v_i$ in the ordering. Hence every pair is counted only from its earlier endpoint, and summing over i gives at least ${\sum }_i(f_i-2)=e\left(Q\right)-2M$ pairs.    □

The bound is tight up to the term $2M$: in a disjoint union of copies of $K_{D,D}$ there are exactly $e\left(Q\right)$ such pairs. This linear behaviour, compared with the quadratic number of pairs joined by paths of length two in a $C_4$-free graph used for $K_4^{*}$ in [5], is the source of the logarithmic loss below.

We use the following results of [5], stated in the notation of this paper; $d\left(\mathsf{\Gamma }\right)$ is the average degree and log the natural logarithm.

(B1)

[5] If a bipartite graph with sides $X,Y$ has $e(X,Y)\le r\left|Y\right|$, then there are $X^{\prime }\subseteq X$, $Y^{\prime }\subseteq Y$ with $|X^{\prime }|\ge \lfloor |X|/(r+1)\rfloor$, $|Y^{\prime }|\ge \lfloor |Y|/(r+1)\rfloor$ and no edges between them.

(B2)

[5] Every graph on $N\ge 12$ vertices has an induced subgraph ${\mathsf{\Gamma }}^{\prime }$ with $\left|V\right({\mathsf{\Gamma }}^{\prime }\left)\right|\ge N/6$ and $\mathsf{\Delta }\left({\mathsf{\Gamma }}^{\prime }\right)\le d\left({\mathsf{\Gamma }}^{\prime }\right)log\left|V\left({\mathsf{\Gamma }}^{\prime }\right)\right|$.

(B3)

[5] Let F have m edges, let A be a set of k vertices of largest degree in F, let $\mathsf{\Gamma }$ have N vertices and average degree d, and let ${\sigma }^{\prime }$ embed $F\left[A\right]$ into $\overline{\mathsf{\Gamma }}$. If $N\ge {\displaystyle \frac{4m}{k+1}}max\left({max}_{v\in {\sigma }^{\prime }\left(A\right)}{d}_{\mathsf{\Gamma }}\left(v\right),2d\right)+2v\left(F\right)$, then ${\sigma }^{\prime }$ extends to an embedding of F into $\overline{\mathsf{\Gamma }}$.

(B4)

[5] If $\mathsf{\Gamma }$ has N vertices and average degree $d\le \sqrt{(N^2-2Nv\left(F\right))/\left(48e\left(F\right)\right)}$, then $\overline{\mathsf{\Gamma }}$ contains F.

(B5)

[5] Let F be bipartite with sides $A,B$ and m edges, let $A^{\prime }$ be a set of k vertices of largest degree in A and $B^{\prime }$ a set of ℓ vertices of largest degree in B, let $\mathsf{\Gamma }$ have N vertices and average degree d, and let ${\sigma }^{\prime }$ embed $F[A^{\prime }\cup B^{\prime }]$ into $\overline{\mathsf{\Gamma }}$. Suppose that $B^{\prime }=B$ or $N\ge {\displaystyle \frac{2m}{\ell +1}}{max}_{v\in {\sigma }^{\prime }\left(A^{\prime }\right)}{d}_{\mathsf{\Gamma }}\left(v\right)+2v\left(F\right)$, and that $A^{\prime }=A$ or $N\ge {\displaystyle \frac{m}{k+1}}max\left({max}_{v\in {\sigma }^{\prime }\left(B^{\prime }\right)}{d}_{\mathsf{\Gamma }}\left(v\right),2d\right)+v\left(F\right)$. Then ${\sigma }^{\prime }$ extends to an embedding of F into $\overline{\mathsf{\Gamma }}$.

(B6)

[5] (a consequence of Shearer’s method [14]; see also [4]) A graph with N vertices, average degree d and at most T triangles has an independent set of size at least $0.1{\displaystyle \frac{N}{d}}\left(logd-{\displaystyle \frac{1}{2}}log(T/N)\right)$.

Lemma 7. 

Every $W_5^{-}$-free graph Γ with N vertices and average degree $d\ge 36$ satisfies $\alpha \left(\mathsf{\Gamma }\right)\ge {\displaystyle \frac{Nlogd}{320d}}$. 

Proof. 

Let W be the set of vertices of degree at most $2d$, so $\left|W\right|\ge N/2$, and let $d_0$ be the average degree of $\mathsf{\Gamma }\left[W\right]$. If $d_0\le d^{7/8}$, then $\alpha \left(\mathsf{\Gamma }\right)\ge \left|W\right|/(d_0+1)\ge N/\left(4d^{7/8}\right)\ge Nlogd/\left(320d\right)$, since $logd\le 80d^{1/8}$. Otherwise, Lemma 5(c) with $D=2d$ bounds the number of triangles of $\mathsf{\Gamma }\left[W\right]$ by $\left|W\right|\left({\displaystyle \frac{3}{2}}{d}_0+{\displaystyle \frac{{\left(2d\right)}^{3/2}}{6}}+{\displaystyle \frac{d}{6}}\right)\le \left|W\right|d^{3/2}$, using $d_0\le 2d$ and $d\ge 36$. Then  gives

$$\alpha \left(\mathsf{\Gamma }\right)\ge 0.1{\displaystyle \frac{\left|W\right|}{d_0}}\left(log{d}_0-{\displaystyle \frac{3}{4}}logd\right)\ge 0.1·{\displaystyle \frac{N/2}{2d}}·{\displaystyle \frac{1}{8}}logd={\displaystyle \frac{Nlogd}{320d}}.$$

   □

Theorem 6. 

There is an absolute constant C such that $R(W_5^{-},F)\le Ce\left(F\right)\sqrt{log\left(e\right(F)+1)}$ for every bipartite graph F without isolated vertices. 

Proof. 

Let F have sides $A,B$ and m edges, so $v\left(F\right)\le 2m$. For m below a suitable absolute constant $m_0$ the statement holds with C large, since $R(W_5^{-},F)\le R(K_6,K_{2m})$. Let $m\ge m_0$, and let ${\mathsf{\Gamma }}_0$ be a $W_5^{-}$-free graph on $N_0=\lceil {10}^{7}m\sqrt{logm}\rceil$ vertices such that $\overline{{\mathsf{\Gamma }}_0}$ does not contain F. By  there is an induced subgraph $\mathsf{\Gamma }$ with $N\ge N_0/6$ vertices, average degree d and $\mathsf{\Delta }\left(\mathsf{\Gamma }\right)\le dlogN$; it is $W_5^{-}$-free and $\overline{\mathsf{\Gamma }}$ does not contain F. By , $d>\sqrt{(N^2-2Nv\left(F\right))/\left(48m\right)}\ge N/\sqrt{96m}$; hence $d\ge 36$ and $logd\ge {\displaystyle \frac{1}{2}}logm$. Choosing $m_0$ large also gives $2\le logN\le 2logm$.

Let S be the set of pairs of distinct vertices with at most three common neighbours. For a vertex x let $S\left(x\right)$ be the set of vertices y with $\{x,y\}\in S$ and $d\left(y\right)\le d\left(x\right)$.

Case 1: $\left|S\right|\ge 2NdlogN$. Since ${\sum }_x\left|S\left(x\right)\right|\ge \left|S\right|$, some x has $\left|S\right(x\left)\right|\ge 2dlogN$, and as $d\left(x\right)\le dlogN$, the set $Y=S\left(x\right)\setminus N\left(x\right)$ has at least $dlogN$ elements. Every $y\in Y$ has at most three neighbours in $X=N\left(x\right)$. By  with $r=3$ there are $X^{\prime }\subseteq X$ and $Y^{\prime }\subseteq Y$ with $|X^{\prime }|\ge \lfloor d\left(x\right)/4\rfloor$, $|Y^{\prime }|\ge \lfloor dlogN/4\rfloor$ and no edges between them. Map the $k=min\left(\right|Y^{\prime }|,|A\left|\right)$ vertices of largest degree in A into $Y^{\prime }$ and the $\ell =min\left(\right|X^{\prime }|,|B\left|\right)$ vertices of largest degree in B into $X^{\prime }$. Vertices of $Y^{\prime }$ have degree at most $d\left(x\right)$ and vertices of $X^{\prime }$ at most $dlogN$. So the two conditions of  hold, since

$${\displaystyle \frac{2m}{\lfloor d\left(x\right)/4\rfloor +1}}d\left(x\right)+4m\le 12m\le N\mathrm{and}{\displaystyle \frac{m}{\lfloor dlogN/4\rfloor +1}}dlogN+2m\le 6m\le N,$$

and $\overline{\mathsf{\Gamma }}$ contains F, a contradiction.

Case 2: $\left|S\right|<2NdlogN$ and $d\ge 4800\sqrt{m}logN$. For a vertex x let $N^{\prime }\left(x\right)$ be the set of neighbours of x of degree at most $d\left(x\right)$, let $d^{\prime }\left(x\right)=\left|N^{\prime }\left(x\right)\right|$, and let $t\left(x\right)$ be the number of pairs of S inside $N^{\prime }\left(x\right)$. Then ${\sum }_x{d}^{\prime }\left(x\right)\ge e\left(\mathsf{\Gamma }\right)=Nd/2$. Also ${\sum }_{x}t\left(x\right)\le 3\left|S\right|$, because the vertices x counting a pair of S are common neighbours of the pair. Hence

$$\sum _x\left[24logN\left(d^{\prime }\left(x\right)-{\displaystyle \frac{d\left(x\right)}{8}}-{\displaystyle \frac{d}{8}}\right)-t\left(x\right)\right]\ge 24logN·{\displaystyle \frac{Nd}{4}}-3\left|S\right|>0,$$

and we fix x with $d^{\prime }\left(x\right)\ge d\left(x\right)/8$, $d^{\prime }\left(x\right)\ge d/8$ and $t\left(x\right)\le 24d^{\prime }\left(x\right)logN$. Let ${\mathsf{\Gamma }}_1=\mathsf{\Gamma }\left[N^{\prime }\left(x\right)\right]$, $M=d^{\prime }\left(x\right)$, and let $d_1$ be the average degree of ${\mathsf{\Gamma }}_1$. By Lemma 5(a), every pair joined by a path with three edges in ${\mathsf{\Gamma }}_1$ lies in S. So Lemma 6 gives $d_{1}M/2-2M\le t\left(x\right)\le 24MlogN$, that is, $d_1\le 48logN+4\le 50logN$.

Let $A_F$ be a set of $\lfloor M/3\rfloor$ vertices of largest degree in F (or $A_F=V\left(F\right)$ if $v\left(F\right)\le M/3$). Since $\sqrt{(M^2-2M|A_F\left|\right)/\left(48m\right)}\ge M/\left(12\sqrt{m}\right)\ge d/\left(96\sqrt{m}\right)\ge 50logN\ge d_1$, statement  embeds $F\left[A_F\right]$ into $\overline{{\mathsf{\Gamma }}_1}$. The images have degree at most $d\left(x\right)\le 8M$ in $\mathsf{\Gamma }$, and $2d\le 16M$, so the condition of  reads ${\displaystyle \frac{4m}{\lfloor M/3\rfloor +1}}·16M+4m\le 196m\le N$. Hence $\overline{\mathsf{\Gamma }}$ contains F, a contradiction.

Case 3: $d<4800\sqrt{m}logN$. By Lemma 7, the set W of vertices of degree at most $2d$ contains an independent set I with $\left|I\right|\ge Nlogd/\left(320d\right)$: the proof of that lemma finds I inside W. Let $k=min\left(\right|I|,v(F\left)\right)$ and embed a set of k vertices of largest degree of F into I. If $k<v\left(F\right)$, the condition of  is implied by

$$N\ge {\displaystyle \frac{4m·320d}{Nlogd}}·2d+4m,$$

and using $d^2<2.4·{10}^{7}m{log}^{2}N\le 9.3·{10}^{7}m{log}^{2}m$ and $logd\ge {\displaystyle \frac{1}{2}}logm$, the first term is at most $4.8·{10}^{11}{m}^{2}logm/N\le 3·{10}^{5}m\sqrt{logm}$. Since $N\ge N_0/6\ge 1.6·{10}^{6}m\sqrt{logm}$, the condition holds and $\overline{\mathsf{\Gamma }}$ contains F, a contradiction.    □

Remark 2 

(Where the logarithm comes from). Case 2 applies only when $d\gtrsim \sqrt{m}logN$, because Lemma 6 bounds the average degree inside $N^{\prime }\left(x\right)$ by $O(logN)$ rather than by $O\left(\sqrt{logN}\right)$ as in the $K_4^{*}$ case. Case 3 reaches only $d\lesssim {\displaystyle \frac{N}{m}}\sqrt{mlogm}$. For $N=\mathsf{\Theta }\left(m\right)$ the two ranges leave a gap of a factor $\sqrt{logm}$, and N has to be raised by this factor to close it. The obstruction is illustrated by neighbourhoods that are blow-ups of graphs of odd girth at least seven. They are $C_5$-free, the number of pairs joined by paths with three edges is only linear in the number of edges, and every edge lies on a 4-cycle of the neighbourhood, hence in at most three triangles of the host by Lemma 5(a). We do not know whether $R(W_5^{-},F)=O\left(e\left(F\right)\right)$ for every bipartite F; Theorem 1 gives this for $F=K_{n,n}$. 

Remark 3 

(Complete graphs). Lemma 7 also gives $R(W_5^{-},K_n)=O(n^{5/2}/{log}^{3/2}n)$. Indeed, let Γ be $W_5^{-}$-free with $\alpha \left(\mathsf{\Gamma }\right)<n$. A 5-cycle in a neighbourhood $N\left(h\right)$ would form, together with h, a copy of $W_5\supseteq W_5^{-}$. So every neighbourhood is $C_5$-free with independence number less than n, and every degree is less than $R(C_5,K_n)\le c{n}^{3/2}/\sqrt{logn}$, by the theorem of Sudakov [17] and of Li and Zang [12]. If the average degree d is at least 36, Lemma 7 gives $n>Nlogd/\left(320d\right)$. Since $x/logx$ is increasing for $x\ge e$, this yields $N<320nD/logD$ with $D=c{n}^{3/2}/\sqrt{logn}$, that is, $N=O(n^{5/2}/{log}^{3/2}n)$. The earlier bound $R(C_5,K_n)=O\left(n^{3/2}\right)$ of Erdős, Faudree, Rousseau and Schelp [8] would give $O(n^{5/2}/logn)$ in the same way. 

In the other direction, let $m_2\left(H\right)$ be the maximum of $(e\left(H^{\prime }\right)-1)/(v\left(H^{\prime }\right)-2)$ over all subgraphs $H^{\prime }\subseteq H$ with at least three vertices. The probabilistic argument of Spencer [16], based on the Lovász local lemma, gives

$$R(H,K_n)=\mathsf{\Omega }\left({(n/logn)}^{m_2\left(H\right)}\right)$$

for every fixed graph H. Since $m_2\left(W_5^{-}\right)=2$, this gives $R(W_5^{-},K_n)=\mathsf{\Omega }(n^2/{log}^{2}n)$. For $K_4^{*}$ the upper bound $O\left(n^{5/2}\right)$ is due to Bradač, Gishboliner and Sudakov [5]. For both graphs Problem #566 predicts $O\left(n^2\right)$. 

7. Discussion and Open Problems

Remark 4 

(Beyond $K_{n,n}$). For bipartite F, Theorem 2 leaves a factor $\sqrt{loge\left(F\right)}$, explained in Remark 2. For $F=K_n$ the best bound we know is $O(n^{5/2}/{log}^{3/2}n)$ (Remark 3), still far from the predicted $O\left(n^2\right)$. Removing the logarithm for bipartite F would require controlling neighbourhoods that are unions of dense bipartite pieces with few triangles per edge. 

Remark 5 

(Minimum degree three). For the prism, $K_{3,3}$ and $Q_3$, any embedding strategy must eventually place a vertex with three prescribed neighbours, and absence of n-holes gives no control over common neighbourhoods of triples. Deciding whether $R(G,K_{n,n})=O\left(n^2\right)$ for the triangular prism seems to be the most natural next question in this direction.