Canonical Ramsey numbers and properly colored cycles

We improve the previous bounds on the so-called unordered Canonical Ramsey numbers, introduced by Richer [D. Richer, Unordered canonical Ramsey numbers, Journal of Combinatorial Theory Series B 80 (2000) 172–177] as a variant of the canonical Ramsey numbers introduced by Erdős and Rado [P. Erdős, R. Rado, A combinatorial theorem, Journal of the London Mathematical Society 25 (4) (1950) 249–255].Then we prove a conjecture raised by Axenovich, Jiang, and Tuza in [M. Axenovich, T. Jiang, Zs. Tuza, Local anti-Ramsey numbers of graphs, Combinatorics, Probability, and Computing 12 (2003) 495–511], showing that for each k≥4k≥4 and large nn, every edge-coloring of KnKn in which at least 256k256k different colors appear at each vertex contains a properly colored cycle of length exactly kk. Here, a cycle is properly colored   if no two incident edges in it have the same color. The bound 256k256k is tight up to a constant factor.