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]. e 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 ≥ 4 and large n , every edge-coloring of K n in which at least 256 k different colors appear at each vertex contains a properly colored cycle of length exactly k . Here, a cycle is properly colored if no two incident edges in it have the same color. The bound 256 k is tight up to a constant factor.