An application of splittable 4-frames to coloring of Kn,n Original Research Article

Axenovich et al. (J. Combin. Theory Ser. B, to appear) considered the problem of the generalized Ramsey theory. In one case, they use the existence of Steiner triple systems, Pippenger and Spencerʹs theorem on hyperedge coloring, and the probabilistic method to show that r′(Kn,n,C4,3)⩽3n/4(1+o(1)), where r′(Kn,n,C4,3) denotes the minimum number of colors to color the edges of Kn,n such that every 4-cycle receives at least either 3 colors or 2 alternating colors. In this short paper, using techniques from combinatorial design theory, we prove that r′(Kn,n,C4,3)⩽(2n/3)+9 for all n. The result is the best possible since r′(Kn,n,C4,3)>⌊2n/3⌋ as shown by Axenovich et al. (J. Combin. Theory Ser. B, to appear).