Given a graph H, the Ramsey number r (H) is the smallest natural number N such that any
two-colouring of the edges of KN contains a monochromatic copy of H. The existence of …

For integers s,t≥2, the Ramsey number r(s,t) denotes the minimum n such that every n-
vertex graph contains a clique of order s or an independent set of order t. In this paper we …

For fixed s 3, we prove that if optimal Ks-free pseudorandom graphs exist, then the Ramsey
number rs; t/is ts1Co. 1/as t! 1. Our method also improves the best lower bounds for rC; …

For 2≤ s< t 2 ≤ s< t, the Erdős–Rogers function fs, t (n) f _ s, t (n) measures how large a K s
K _s‐free induced subgraph there must be in a K t K _t‐free graph on nn vertices. There has …

The classical hypergraph Ramsey number rk (s, n) is the minimum N such that for every red-
blue coloring of the k-tuples of {1,…, N}, there are s integers such that every k-tuple among …

For an integer $ n\geq 1$, the Erd\H {o} s-Rogers function $ f_ {s}(n) $ is the maximum
integer $ m $ such that every $ n $-vertex $ K_ {s+ 1} $-free graph has a $ K_s $-free …

For fixed $ s\ge 3$, we prove that if optimal $ K_s $-free pseudorandom graphs exist, then
the Ramsey number $ r (s, t)= t^{s-1+ o (1)} $ as $ t\rightarrow\infty $. Our method also …

An r-quasiplanar graph is a graph drawn in the plane with no r pairwise crossing edges. Let
s≥ 3 be an integer and r= 2 s. We prove that there is a constant C such that every r …

We prove several results from different areas of extremal combinatorics, giving complete or
partial solutions to a number of open problems. These results, coming from areas such as …

The Erd\H {o} s-Rogers function $ f_ {s, t} $ measures how large a $ K_s $-free induced
subgraph there must be in a $ K_t $-free graph on $ n $ vertices. While good estimates for …