Coloring the cube with rainbow cycles

Abstract

For every even positive integer k≥4 let f(n,k) denote the minimim number of colors required to color the edges of the n-dimensional cube Qn, so that the edges of every copy of the k-cycle Ck receive k distinct colors. Faudree, Gyárfás, Lesniak and Schelp proved that f(n,4)=n for n=4 or n>5. We consider larger k and prove that if k≡0 (mod 4), then there are positive constants c1,c2 depending only on k such that Our upper bound uses an old construction of Bose and Chowla of generalized Sidon sets. For k≡2 (mod 4), the situation seems more complicated. For the smallest case k=6 we show that with the lower bound holding for n≥3. The upper bound is obtained from Behrend's construction of a subset of integers with no three term arithmetic progression.