Three-colour bipartite Ramsey number for graphs with small bandwidth

We estimate the $3$-colour bipartite Ramsey number for balanced bipartite graphs $H$ with small bandwidth and bounded maximum degree. More precisely, we show that the minimum value of $N$ such that in any $3$-edge colouring of $K_{N,N}$ there is a monochromatic copy of $H$ is at most $\big(3/2+o(1)\big)|V(H)|$. In particular, we determine asymptotically the $3$-colour bipartite Ramsey number for balanced grid graphs.Comment: 15 page

Ramsey numbers of uniform loose paths and cycles

Recently, determining the Ramsey numbers of loose paths and cycles in uniform hypergraphs has received considerable attention. It has been shown that the $2$-color Ramsey number of a $k$-uniform loose cycle $\mathcal{C}^k_n$, $R(\mathcal{C}^k_n,\mathcal{C}^k_n)$, is asymptotically $\frac{1}{2}(2k-1)n$. Here we conjecture that for any $n\geq m\geq 3$ and $k\geq 3,$ $$R(\mathcal{P}^k_n,\mathcal{P}^k_m)=R(\mathcal{P}^k_n,\mathcal{C}^k_m)=R(\mathcal{C}^k_n,\mathcal{C}^k_m)+1=(k-1)n+\lfloor\frac{m+1}{2}\rfloor.$$ Recently the case $k=3$ is proved by the authors. In this paper, first we show that this conjecture is true for $k=3$ with a much shorter proof. Then, we show that for fixed $m\geq 3$ and $k\geq 4$ the conjecture is equivalent to (only) the last equality for any $2m\geq n\geq m\geq 3$. Consequently, the proof for $m=3$ follows

Ramsey numbers of 3-uniform loose paths and loose cycles

Haxell et. al. [%P. Haxell, T. Luczak, Y. Peng, V. R\"{o}dl, A. %Ruci\'{n}ski, M. Simonovits, J. Skokan, The Ramsey number for hypergraph cycles I, J. Combin. Theory, Ser. A, 113 (2006), 67-83] proved that the 2-color Ramsey number of 3-uniform loose cycles on $2n$ vertices is asymptotically $\frac{5n}{2}$. Their proof is based on the method of Regularity Lemma. Here, without using this method, we generalize their result by determining the exact values of 2-color Ramsey numbers involving loose paths and cycles in 3-uniform hypergraphs. More precisely, we prove that for every $n\geq m\geq 3$, $R(\mathcal{P}^3_n,\mathcal{P}^3_m)=R(\mathcal{P}^3_n,\mathcal{C}^3_m)=R(\mathcal{C}^3_n,\mathcal{C}^3_m)+1=2n+\lfloor\frac{m+1}{2}\rfloor$ and for $n>m\geq3$, $R(\mathcal{P}^3_m,\mathcal{C}^3_n)=2n+\lfloor\frac{m-1}{2}\rfloor$. These give a positive answer to a question of Gy\'{a}rf\'{a}s and Raeisi [The Ramsey number of loose triangles and quadrangles in hypergraphs, Electron. J. Combin. 19 (2012), #R30]

The Size-Ramsey Number of 3-uniform Tight Paths

Given a hypergraph H, the size-Ramsey number ˆr2(H) is the smallest integer m such that there exists a hypergraph G with m edges with the property that in any colouring of the edges of G with two colours there is a monochromatic copy of H. We prove that the size-Ramsey number of the 3-uniform tight path on n vertices Pn(3) is linear in n, i.e., ˆr2(Pn(3)) = O(n). This answers a question by Dudek, La Fleur, Mubayi, and Rödl for 3-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417–434], who proved ˆr2(Pn(3)) = O(n3/2 log3/2 n)

Hypergraph Ramsey numbers of cliques versus stars

Let $K_m^{(3)}$ denote the complete $3$-uniform hypergraph on $m$ vertices and $S_n^{(3)}$ the $3$-uniform hypergraph on $n+1$ vertices consisting of all $\binom{n}{2}$ edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off-diagonal Ramsey number $r(K_{4}^{(3)},S_n^{(3)})$ exhibits an unusual intermediate growth rate, namely, $$2^{c \log^2 n} \le r(K_{4}^{(3)},S_n^{(3)}) \le 2^{c' n^{2/3}\log n}$$ for some positive constants $c$ and $c'$. The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum $N$ such that any $2$-edge-coloring of the Cartesian product $K_N \square K_N$ contains either a red rectangle or a blue $K_n$?Comment: 13 page

Tur\'an's problem and Ramsey numbers for trees

Let $T_n^1=(V,E_1)$ and $T_n^2=(V,E_2)$ be the trees on $n$ vertices with $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_1=\{v_0v_1,\ldots,v_0v_{n-3},v_{n-4}v_{n-2},v_{n-3}v_{n-1}\}$, and $E_2=\{v_0v_1,\ldots,$ $v_0v_{n-3},v_{n-3}v_{n-2}, v_{n-3}v_{n-1}\}$. In this paper, for $p\ge n\ge 5$ we obtain explicit formulas for \ex(p;T_n^1) and \ex(p;T_n^2), where \ex(p;L) denotes the maximal number of edges in a graph of order $p$ not containing $L$ as a subgraph. Let r(G\sb 1, G\sb 2) be the Ramsey number of the two graphs $G_1$ and $G_2$. In this paper we also obtain some explicit formulas for $r(T_m,T_n^i)$, where $i\in\{1,2\}$ and $T_m$ is a tree on $m$ vertices with $\Delta(T_m)\le m-3$.Comment: 21 page

Improved bounds for the Ramsey number of tight cycles versus cliques

The 3-uniform tight cycle $C_s^3$ has vertex set $Z_s$ and edge set $\{\{i, i+1, i+2\}: i \in Z_s\}$. We prove that for every $s \not\equiv 0$ (mod 3) and $s \ge 16$ or $s \in \{8,11,14\}$ there is a $c_s>0$ such that the 3-uniform hypergraph Ramsey number $$r(C_s^3, K_n^3)< 2^{c_s n \log n}$$ This answers in strong form a question of the author and R\"odl who asked for an upper bound of the form $2^{n^{1+\epsilon_s}}$ for each fixed $s \ge 4$, where $\epsilon_s \rightarrow 0$ as $s \rightarrow \infty$ and $n$ is sufficiently large. The result is nearly tight as the lower bound is known to be exponential in $n$

Hypergraph Ramsey numbers

The Ramsey number r_k(s,n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all k-tuples from this set are red (blue). In this paper we obtain new estimates for several basic hypergraph Ramsey problems. We give a new upper bound for r_k(s,n) for k \geq 3 and s fixed. In particular, we show that r_3(s,n) \leq 2^{n^{s-2}\log n}, which improves by a factor of n^{s-2}/ polylog n the exponent of the previous upper bound of Erdos and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there are constants c_1,c_2>0 such that r_3(s,n) \geq 2^{c_1 sn \log (n/s)} for all 4 \leq s \leq c_2n. When s is a constant, it gives the first superexponential lower bound for r_3(s,n), answering an open question posed by Erdos and Hajnal in 1972. Next, we consider the 3-color Ramsey number r_3(n,n,n), which is the minimum N such that every 3-coloring of the triples of an N-element set contains a monochromatic set of size n. Improving another old result of Erdos and Hajnal, we show that r_3(n,n,n) \geq 2^{n^{c \log n}}. Finally, we make some progress on related hypergraph Ramsey-type problems

Path-fan Ramsey numbers

For two given graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest positive integer $p$ such that for every graph $F$ on $p$ vertices the following holds: either $F$ contains $G$ as a subgraph or the complement of $F$ contains $H$ as a subgraph. In this paper, we study the Ramsey numbers $R(P_{n},F_{m})$, where $P_{n}$ is a path on $n$ vertices and $F_{m}$ is the graph obtained from $m$ disjoint triangles by identifying precisely one vertex of every triangle ($F_{m}$ is the join of $K_{1}$ and $mK_{2}$). We determine exact values for $R(P_{n},F_{m})$ for the following values of $n$ and $m$: $n=1,2$ or $3$ and $m\geq 2$; $n\geq 4$ and $2\leq m\leq (n+1)/2$; $n\geq7$ and $m=n-1$ or $m=n$; $n\geq 8$ and $(k\cdot n-2k+1)/2\leq m\leq (k\cdot n-k+2)/2$ with $3\leq k\leq n-5$; $n=4,5$ or $6$ and $m\geq n-1$; $n\geq 7$ and $m\geq (n-3)^2/2$. We conjecture that $R(P_{n},F_{m}) \leq 2m+n-3$ for the other values of $m$ and $n$. \u

Three colour bipartite Ramsey number of cycles and paths

The $k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $n$ for which every $k$-edge-coloured complete bipartite graph $K_{n,n}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey numbers was initiated, over 40 years ago, by Faudree and Schelp and, independently, by Gy\'arf\'as and Lehel, who determined the $2$-colour Ramsey number of paths. In this paper we determine asymptotically the $3$-colour bipartite Ramsey number of paths and (even) cycles.Comment: 15 pages, 3 figure