Design theory
Frames for Kirkman triple systems
Discrete Mathematics
Asymptotic behavior of the chromatic index for hypergraphs
Journal of Combinatorial Theory Series A
Linear spaces with many small lines
Proceedings of the first International conference on Linear spaces
On generalized Ramsey theory: The bipartite case
Journal of Combinatorial Theory Series B
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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 Spencer's theorem on hyperedge coloring, and the probabilistic method to show that r'(Kn,n, C4, 3) ≤ 3n/4(1 + o(1)), wherer'(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).