In 1983, Burr and Erdős initiated the study of Ramsey goodness problems.Nikiforov and Rousseau (2009) resolved almost all goodness questions raised by Burr and Erdős, in which the bounds on the parameters are of tower type since their proofs rely on the regularity lemma. Let $B_{k,n}$ be the book graph on $n$ vertices which consists of $n-k$ copies of $K_{k+1}$ all sharing a common $K_k$, and let $H=K_p(a_1,\dots,a_{p})$ be the complete $p$-partite graph with parts of sizes $a_1,\dots,a_{p}$.
Recently, avoiding use of the regularity lemma, Fox, He and Wigderson (2021) revisit several Ramsey goodness results involving books. They comment that it would be very interesting to see how far one can push these ideas. In particular, they conjecture that for all integers $k, p, t\ge 2$, there exists some $δ>0$ such that for all $n\ge 1$, $1\leq a_1\le\cdots\le a_{p-1}\le t$ and $a_p \le δn$, we have $r(H, B_{k,n})= (p-1)(n-1)+d_k(n,K_{a_1,a_2})+1,$ where $d_k(n,K_{a_1,a_2})$ is the maximum $d$ for which there is an $(n+d-1)$-vertex $K_{a_1,a_2}$-free graph in which at most $k-1$ vertices have degree less than $d$.They verify the conjecture when $a_1=a_2=1$.
Building upon the work of Fox et al. (2021), we make a substantial step by showing that the conjecture "roughly" holds if $a_1=1$ and $a_2|(n-1-k)$, i.e. $a_2$ divides $n-1-k$. Moreover, avoiding use of the regularity lemma, we prove that for every $k, a\geq 1$ and $p\ge2$, there exists $δ>0$ such that for all large $n$ and $b\le δ\ln n$, $r(K_p(1,a,b,\dots,b), B_{k,n})= (p-1)(n-1)+k(p-1)(a-1)+1$ if $a|(n-1-k)$, where the case when $a=1$ has been proved by Nikiforov and Rousseau (2009) using the regularity lemma. The bounds on $1/δ$ we obtain are not of tower-type since our proofs do not rely on the regularity lemma.
In a seminal paper from 1983, Burr and Erdos started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemeredi regularity lemma, embedding of sparse graphs, Turan type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving all but one of the Burr-Erdos problems.
A book of size q is the union of q triangles sharing a common edge. We find the exact Ramsey number of books of size q versus books of size p when p<q/6-o(q).
A book of size N is the union of N triangles sharing a common edge. We show that the Ramsey number of a book of size N vs. a book of size M equals 2N+3 for all N>(10^6)M. Our proof is based on counting.
Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If graph $G$ is connected, it is well known and easy to show that $R(G,H) \geq (|G|-1)(χ(H)-1)+σ(H)$, where $χ(H)$ is the chromatic number of $H$ and $σ$ the size of the smallest color class in a $χ(H)$-coloring of $H$. A graph $G$ is called $H$-good if $R(G,H)= (|G|-1)(χ(H)-1)+σ(H)$. The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively studied si…
Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it is well known and easy to show that $R(G,H) \geq (|G|-1)(χ(H)-1)+σ(H)$, where $χ(H)$ is the chromatic number of $H$ and $σ(H)$ is the size of the smallest color class in a $χ(H)$-coloring of $H$. A graph $G$ is called $H$-good if $R(G,H)= (|G|-1)(χ(H)-1)+σ(H)$. The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively st…
Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for books $B_{m} = K_{2} + \overline{K_{m}}$ and prove sharp results for $m \leq 5$.
Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it is well known and easy to show that $R(G,H) \geq (|G|-1)(χ(H)-1)+σ(H)$, where $χ(H)$ is the chromatic number of $H$ and $σ(H)$ is the size of the smallest color class in a $χ(H)$-coloring of $H$. A graph $G$ is called $H$-good if $R(G,H)= (|G|-1)(χ(H)-1)+σ(H)$. The notion of Ramsey goodness was introduced by Burr and Erdős in 1983 and has been extensively st…
We say that a graph $G$ is Ramsey for $H_1$ versus $H_2$, and write $G \to (H_1,H_2)$, if every red-blue colouring of the edges of $G$ contains either a red copy of $H_1$ or a blue copy of $H_2$. In this paper we study the threshold for the event that the Erdős--Rényi random graph $G(N,p)$ is Ramsey for a clique versus a path. We show that $$G\big( (1 + \varepsilon) rn,p \big) \to (K_{r+1},P_n)$$ with high probability if $p \gg n^{-2 / (r + 1)}$, and $$G\big( rn + t, p \big) \to (K_{r+1},P_n)$$ with high probability if $p \gg n^{-2 / (r + 2)}$ and $t \gg 1/p$. Both of these results are sharp (…
The book graph $B_n^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$. The Ramsey numbers of $B_n^{(k)}$ are known to have strong connections to the classical Ramsey numbers of cliques. Recently, the first author determined the asymptotic order of these Ramsey numbers for fixed $k$, thus answering an old question of Erdős, Faudree, Rousseau, and Schelp. In this paper, we first provide a simpler proof of this theorem. Next, answering a question of the first author, we present a different proof that avoids the use of Szemerédi's regularity lemma, thus providing much tighter …
A book $B_n$ is a graph which consists of $n$ triangles sharing a common edge. In 1978, Rousseau and Sheehan conjectured that the Ramsey number satisfies $r(B_m,B_n)\le 2(m+n)+c$ for some constant $c>0$. In this paper, we obtain that $r(B_m, B_n)\le 2(m+n)+o(n)$ for all $m\le n$ and $n$ large, which confirms the conjecture of Rousseau and Sheehan asymptotically. As a corollary, our result implies that a related conjecture of Faudree, Rousseau and Sheehan (1982) on strongly regular graph holds asymptotically.
The book graph $B_n^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$. In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n^{(k)}, B_n^{(k)})$. Here we consider the natural off-diagonal variant $r(B_{cn}^{(k)}, B_n^{(k)})$ for fixed $c \in (0,1]$. In this more general setting, we show that an interesting dichotomy emerges: for very small $c$, a simple $k$-partite construction dictates the Ramsey function and all nearly-extremal colorings are close to being $k$-partite, while, for $c$ bounded away from $0$, random colorings of an appropriate density ar…
We show that in every two-colouring of the edges of the complete graph $K_N$ there is a monochromatic $K_k$ which can be extended in at least $(1 + o_k(1))2^{-k}N$ ways to a monochromatic $K_{k+1}$. This result is asymptotically best possible, as may be seen by considering a random colouring. Equivalently, defining the book $B_n^{(k)}$ to be the graph consisting of $n$ copies of $K_{k+1}$ all sharing a common $K_k$, we show that the Ramsey number $r(B_n^{(k)}) = 2^k n + o_k(n)$. In this form, our result answers a question of Erdős, Faudree, Rousseau and Schelp and establishes an asymptotic ver…
Let $B_n^{(k)}$ be the book graph which consists of $n$ copies of $K_{k+1}$ all sharing a common $K_k$, and let $C_m$ be a cycle of length $m$. In this paper, we first determine the exact value of $r(B_n^{(2)}, C_m)$ for $\frac{8}{9}n+112\le m\le \lceil\frac{3n}{2}\rceil+1$ and $n \geq 1000$. This answers a question of Faudree, Rousseau and Sheehan (Cycle--book Ramsey numbers, {\it Ars Combin.,} {\bf 31} (1991), 239--248) in a stronger form when $m$ and $n$ are large. Building upon this exact result, we are able to determine the asymptotic value of $r(B_n^{(k)}, C_n)$ for each $k \geq 3$. Name…
An r-book of size q is a union of q (r+1)-cliques sharing a common r-clique. We find exactly the Ramsey number of a p-clique versus r-books of sufficiently large size. Furthermore, we find asymptotically the Ramsey number of any fixed p-chromatic graph versus r-books of sufficiently large size. The key element in our proofs is Szemeredi's Regularity Lemma.
A book $B_n$ is a graph which consists of $n$ triangles sharing a common edge. In this paper, we study Ramsey numbers of quadrilateral versus books. Previous results give the exact value of $r(C_4,B_n)$ for $1\le n\le 14$. We aim to show the exact value of $r(C_4,B_n)$ for infinitely many $n$. To achieve this, we first prove that $r(C_4,B_{(m-1)^2+(t-2)})\le m^2+t$ for $m\ge4$ and $0 \leq t \leq m-1$. This improves upon a result by Faudree, Rousseau and Sheehan (1978) which states that \begin{align*} r(C_4,B_n)\le g(g(n)), \;\;\text{where}\;\;g(n)=n+\lfloor\sqrt{n-1}\rfloor+2. \end{align*} Com…
For a graph $G$, we write $G\rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$ if every blue-red colouring of the edges of $G$ contains either a blue copy of $K_{r+1}$, or a red copy of each tree with $n$ edges and maximum degree at most $D$. In 1977, Chvátal proved that for any integers $r,n,D \ge 2$, $K_N \rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$ if and only if $N \ge rn+1$. We prove a random analogue of Chvátal's theorem for bounded degree trees, that is, we show that for each $r,D\ge 2$ there exist constants $C,C'>0$ such that if $p \ge C{n}^{-2/(r+2)}$ and $N \geq rn + C'/p$, then \[G(…
A celebrated result of Chvátal, Rödl, Szemerédi and Trotter states (in slightly weakened form) that, for every natural number $Δ$, there is a constant $r_Δ$ such that, for any connected $n$-vertex graph $G$ with maximum degree $Δ$, the Ramsey number $R(G,G)$ is at most $r_Δn$, provided $n$ is sufficiently large. In 1987, Burr made a strong conjecture implying that one may take $r_Δ= Δ$. However, Graham, Rödl and Ruciński showed, by taking $G$ to be a suitable expander graph, that necessarily $r_Δ> 2^{cΔ}$ for some constant $c>0$. We show that the use of expanders is essential: if we impose the…
The size Ramsey number of a graph $H$ is defined as the minimum number of edges in a graph $G$ such that there is a monochromatic copy of $H$ in every two-coloring of $E(G)$. The size Ramsey number was introduced by Erdős, Faudree, Rousseau, and Schelp in 1978 and they ended their foundational paper by asking whether one can determine up to a constant factor the size Ramsey numbers of three families of graphs: complete bipartite graphs, book graphs (obtained by adding many common neighbors to the vertices of a clique), and starburst graphs (obtained by adding many pendant edges to each vertex …
For two graphs $G^<$ and $H^<$ with linearly ordered vertex sets, the ordered Ramsey number $r_<(G^<,H^<)$ is the minimum $N$ such that every red-blue coloring of the edges of the ordered complete graph on $N$ vertices contains a red copy of $G^<$ or a blue copy of $H^<$. For a positive integer $n$, a nested matching $NM^<_n$ is the ordered graph on $2n$ vertices with edges $\{i,2n-i+1\}$ for every $i=1,\dots,n$. We improve bounds on the ordered Ramsey numbers $r_<(NM^<_n,K^<_3)$ obtained by Rohatgi, we disprove his conjecture by showing $4n+1 \leq r_<(NM^<_n,K^<_3) \leq (3+\sqrt{5})n$ for eve…
