Ramsey Number of a Connected Triangle Matching

We determine the 2-color Ramsey number of a connected triangle matching c(nK3) which is a graph with n vertex disjoint triangles plus (at least n − 1) further edges that keep these triangles connected. We obtain that R(c(nK3), c(nK3)) = 7n − 2, somewhat larger than in the classical result of Burr, Erdős and Spencer for a triangle matching, R(nK3, nK3) = 5n. The motivation is to determine the Ramsey number R(C2 n, C 2 n) of the square of a cycle C2 n. We apply our Ramsey result for connected triangle matchings to show that the Ramsey number of an “almost” square of a cycle C n (a cycle of length n in which all but at most a constant c of short diagonals are present) is asymptotic to 7n/3.