Related papers: Nearly tight exponents for off-diagonal Ramsey num…

The Ramsey number $r(t;\ell)$ is the smallest $n$ such that every $\ell$-coloring of the edges of $K_n$ gives a monochromatic $K_{t}$. In recent years, there have been several improvements on asymptotic lower bounds for these numbers when…

The Ramsey number $r(G)$ of a graph $G$ is the smallest integer $n$ such that any $2$ colouring of the edges of a clique on $n$ vertices contains a monochromatic copy of $G$. Determining the Ramsey number of $G$ is a central problem of…

An ordered hypergraph is a hypergraph $H$ with a specified linear ordering of the vertices, and the appearance of an ordered hypergraph $G$ in $H$ must respect the specified order on $V(G)$. In on-line Ramsey theory, Builder iteratively…

Chv\'atal showed that for any tree $T$ with $k$ edges the Ramsey number $R(T,n)=k(n-1)+1$ ("Tree-complete graph Ramsey numbers." Journal of Graph Theory 1.1 (1977): 93-93). For $r=3$ or $4$, we show that, if $T$ is an $r$-uniform…

Given a hypergraph $G$ and a subhypergraph $H$ of $G$, the \emph{odd Ramsey number} $r_{odd}(G,H)$ is the minimum number of colors needed to edge-color $G$ so that every copy of $H$ intersects some color class in an odd number of edges.…

A k-ary semi-algebraic relation E on R^d is a subset of R^{kd}, the set of k-tuples of points in R^d, which is determined by a finite number of polynomial equations and inequalities in kd real variables. The description complexity of such a…

Using cyclic graphs I give new lower bounds for two color and multicolor Ramsey numbers: R(4,16)>163, R(5,11)>170, R(5,12)>190, R(5,13)>212, R(5,14)>238, R(3,3,9)>117, R(3,3,10)>141 and R(3,3,11)>157. Improving the previous best known…

Let $r(G,H)$ be the smallest integer $N$ such that for any $2$-coloring (say, red and blue) of the edges of $K\_n$, $n\geqslant N$, there is either a red copy of $G$ or a blue copy of $H$. Let $K\_n-K\_{1,s}$ be the complete graph on $n$…

A celebrated result of Chv\'atal, R\"odl, Szemer\'edi and Trotter states (in slightly weakened form) that, for every natural number $\Delta$, there is a constant $r_\Delta$ such that, for any connected $n$-vertex graph $G$ with maximum…

In this work, we give several new upper and lower bounds on Ramsey numbers for books and wheels, including a tight upper bound establishing $R(W_5, W_7) = 15$, matching upper and lower bounds giving $R(W_5, W_9) = 18$, $R(B_2, B_8) = 21$,…

In this paper, we prove that for every $k$ and every graph $H$ with $m$ edges and no isolated vertices, the Ramsey number $R(C_k,H)$ is at most $2m+\lfloor \frac{k-1}{2} \rfloor$, provided $m$ is sufficiently large with respect to $k$. This…

A set of vertices $X\subseteq V$ in a simple graph $G(V,E)$ is irredundant if each vertex $x\in X$ is either isolated in the induced subgraph $G[X]$ or else has a private neighbor $y\in V\setminus X$ that is adjacent to $x$ and to no other…

Let $r(k)$ denote the maximum number of edges in a $k$-uniform intersecting family with covering number $k$. Erd\H{o}s and Lov\'asz proved that $ \lfloor k! (e-1) \rfloor \leq r(k) \leq k^k.$ Frankl, Ota, and Tokushige improved the lower…

Ramsey theory is the study of conditions under which mathematical objects show order when partitioned. Ramsey theory on the integers concerns itself with partitions of $[1,n]$ into $r$ subsets and asks the question whether one (or more) of…

Let $f(n,p,q)$ denote the minimum number of colors needed to color the edges of $K_n$ so that every copy of $K_p$ receives at least $q$ distinct colors. In this note, we show $\frac{6}{7}(n-1) \leq f(n,5,8) \leq n + o(n)$. The upper bound…

Extending the concept of Ramsey numbers, Erd{\H o}s and Rogers introduced the following function. For given integers $2\le s<t$ let $$ f_{s,t}(n)=\min \{\max \{|W| : W\subseteq V(G) {and} G[W] {contains no} K_s\} \}, $$ where the minimum is…

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…

For two graphs $G$ and $H$, let $r(G,H)$ and $r_*(G,H)$ denote the Ramsey number and star-critical Ramsey number of $G$ versus $H$, respectively. In 1996, Li and Rousseau proved that $r(K_{m},F_{t,n})=tn(m-1)+1$ for $m\geq 3$ and…

A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph of G…

We improve the upper bound on the Ramsey number $R(5,5)$ from $R(5,5) \le 49$ to $R(5,5) \le 48$. We also complete the catalogue of extremal graphs for $R(4,5)$.