Graphs with asymmetric Ramsey properties

TL;DR

The paper constructs graphs $G$ with $G \nrightarrow (K_k,K_k)$ but $G \rightarrow (K_{R(k)-1},K_{k-1})$ for every $k\ge3$ by combining probabilistic methods with the hypergraph container framework. It shows that a carefully chosen random $s$-uniform hypergraph with $s=R(k)-1$ yields a primal graph $G[\mathcal{H}_0]$ that is Ramsey for $(K_s,K_{k-1})$ under high density, while a linear $k$-conformal subhypergraph allows a coloring avoiding any monochromatic $K_k$, so $G[\mathcal{H}_0] \nrightarrow K_k$. The results extend Nešetřil–Rödl's classical theorem into an asymmetric setting and illuminate Ramsey-equivalence questions, providing both probabilistic constructions and pathways to potential deterministic analogues.

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 Nešetřil and Rödl [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 $G\rightarrow(K_{k-1}, K_{k-1})$.

Theorems & Definitions (18)

• Theorem 1.1: Nešetřil & Rödl, 1976
• Theorem 1.2
• Lemma 2.1: Container Lemma
• proof
• Theorem 2.4
• proof
• Claim 2.5
• proof : Proof of the claim
• Theorem 3.1
• proof : Proof of Theorem \ref{['lemma:cored']}