Ramsey number of a connected triangle matching

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 $R(C_n<sup>2,C_n<sup>2)$ of the square of a cycle $C_n<sup>2$. We apply our Ramsey result for connected triangle matchings to show that the Ramsey number of an "almost" square of a cycle $C_n<sup>{2,c}$ (a cycle of length $n$ in which all but at most a constant number $c$ of short diagonals are present) is asymptotic to $7n/3$.