Ramsey number of a connected triangle matching

TL;DR

This paper determines the 2-color Ramsey number for connected triangle matchings, showing it is 7n-2, and applies this to approximate the Ramsey number of cycle squares with almost all diagonals present.

Contribution

It provides the exact Ramsey number for connected triangle matchings and applies this to approximate the Ramsey number of cycle squares with missing diagonals.

Findings

Ramsey number for connected triangle matching is 7n-2.

Approximate Ramsey number for almost cycle squares is 7n/3.

Extension of classical triangle matching results to connected matchings.

Abstract

We determine the $2$-color Ramsey number of a {\em connected} triangle matching $c(nK_3)$ which is any connected graph containing $n$ vertex disjoint triangles. We obtain that $R(c(nK_3),c(nK_3))=7n-2$, somewhat larger than in the classical result of Burr, Erd\H os and Spencer for a triangle matching, $R(nK_3,nK_3)=5n$. The motivation is to determine the Ramsey number $R(C_n^2,C_n^2)$ of the square of a cycle $C_n^2$. We apply our Ramsey result for connected triangle matchings to show that the Ramsey number of an "almost" square of a cycle $C_n^{2,c}$ (a cycle of length $n$ in which all but at most a constant number $c$ of short diagonals are present) is asymptotic to $7n/3$.