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 …

Extremal combinatorics is an area in discrete mathematics that has developed spectacularly
during the last decades. This paper contains a collection of problems and results in the area …

We confirm a conjecture of Fox, Pach, and Suk, that for every $ d> 0$, there exists $ c> 0$
such that every $ n $-vertex graph of VC-dimension at most $ d $ has a clique or stable set …

Abstract The Erdös–Hajnal conjecture states that for every graph H, there exists a constant
such that every graph G with no induced subgraph isomorphic to H has either a clique or a …

It is known that families of graphs with a semialgebraic edge relation of bounded complexity
satisfy much stronger regularity properties than arbitrary graphs, and can be decomposed …

We prove that, for every family F of n semi-algebraic sets in Rd of constant description
complexity, there exist a positive constant ɛ that depends on the maximum complexity of the …

Erd\H {o} s and Hajnal conjectured that for every graph $ H $, there exists $ c> 0$ such that
every $ H $-free graph $ G $ has a clique or a stable set of size at least $| G|^ c $(" $ H …

In this paper, we present several density-type theorems which show how to find a copy of a
sparse bipartite graph in a graph of positive density. Our results imply several new bounds …

A tournament is a complete graph with its edges directed, and colouring a tournament
means partitioning its vertex set into transitive subtournaments. For some tournaments H …

Abstract The Vapnik–Chervonenkis dimension (in short, VC-dimension) of a graph is
defined as the VC-dimension of the set system induced by the neighborhoods of its vertices …