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

AbstractAxenovich 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 Spencer's 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)

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs