Graphs with asymmetric Ramsey properties

TL;DR

This paper constructs graphs demonstrating asymmetric Ramsey properties, showing that for each k, there exists a graph that avoids certain monochromatic cliques yet guarantees others, extending classical symmetric results.

Contribution

The authors prove the existence of graphs with asymmetric Ramsey properties using probabilistic methods and hypergraph containers, generalizing classical symmetric theorems.

Findings

Existence of graphs avoiding certain monochromatic cliques while guaranteeing others.

Extension of classical symmetric Ramsey theorems to asymmetric cases.

Application of probabilistic and hypergraph container techniques.

Abstract

Given positive integers $k$ and $\ell$ we write $G \rightarrow (K_k,K_\ell)$ if every 2-colouring of the edges of $G$ yields a red copy of $K_k$ or a blue copy of $K_\ell$ and we denote by $R(k)$ the minimum $n$ such that $K_n\rightarrow (K_k,K_k)$. By using probabilistic methods and hypergraph containers we prove that for every integer $k \geq 3$, there exists a graph $G$ such that $G \nrightarrow (K_k,K_k)$ and $G \rightarrow (K_{R(k)-1},K_{k-1})$. This result can be viewed as a variation of a classical theorem of Ne\v{s}et\v{r}il and R\"odl [The Ramsey property for graphs with forbidden complete subgraphs, Journal of Combinatorial Theory, Series B, 20 (1976), 243-249], who proved that for every integer $k\geq 2$ there exists a graph $G$ with no copies of $K_k$ such that $G\rightarrow(K_{k-1}, K_{k-1})$.