Infinitely many minimally non-Ramsey size-linear graphs

Yuval Wigderson

Infinitely many minimally non-Ramsey size-linear graphs

Yuval Wigderson

Abstract

A graph $G$ is said to be Ramsey size-linear if $r(G,H) =O_G (e(H))$ for every graph $H$ with no isolated vertices. Erdős, Faudree, Rousseau, and Schelp observed that $K_4$ is not Ramsey size-linear, but each of its proper subgraphs is, and they asked whether there exist infinitely many such graphs. In this short note, we answer this question in the affirmative.

Infinitely many minimally non-Ramsey size-linear graphs

Abstract

A graph

is said to be Ramsey size-linear if

for every graph

with no isolated vertices. Erdős, Faudree, Rousseau, and Schelp observed that

is not Ramsey size-linear, but each of its proper subgraphs is, and they asked whether there exist infinitely many such graphs. In this short note, we answer this question in the affirmative.

Paper Structure (4 theorems)

This paper contains 4 theorems.

Key Result

Theorem 1

There exist infinitely many graphs $G$ which are not Ramsey size-linear, but every proper subgraph $G' \subsetneq G$ is Ramsey size-linear.

Theorems & Definitions (5)

Theorem 1
Lemma 2
Lemma 3: MR1264714
Lemma 4
proof : Proof of \ref{['thm:main']}