Monochromatic even cycles

We prove that any r-coloring of the edges of Km contains a monochromatic even cycle, where m = 3r + 1 if r is odd and m = 3r if r is even. We also prove that Km−1 has an r-coloring without monochromatic even cycles. An easy exercise, perhaps folkloristic, says that in any r-coloring of the edges of K2r+1 there is a monochromatic odd cycle (and this is not true for K2r). This note explores what happens if we ask the same question for even cycles. Let f(r) denote the smallest integer m for which there is a monochromatic even cycle in every edge coloring of Km. Theorem 1. For odd r, f(r) = 3r + 1 and for even r, f(r) = 3r. Every graph with n vertices and with more than m = b3(n− 1)/2c edges contains a Θ-graph, i.e. three internally vertex disjoint paths connecting the same pair of vertices (see [1], Exercise 10.1). Since a Θ-graph obviously contains an even cycle, any graph with n vertices and more than m edges contains an even cycle. This easily implies that the stated values are upper bounds of f(r) in Theorem 1. Indeed, considering the majority color, one can easily check that⌈( 3r+1 2 ) r ⌉ > ⌊ 3(3r) 2 ⌋ if r is odd