The Erdős–Gyárfás function with respect to Gallai‐colorings

Abstract

AbstractFor fixed and , an edge‐coloring of the complete graph is said to be a ‐coloring if every receives at least distinct colors. The function is the minimum number of colors needed for to have a ‐coloring. This function was introduced about 45 years ago, but was studied systematically by Erdős and Gyárfás in 1997, and is now known as the Erdős–Gyárfás function. In this paper, we study with respect to Gallai‐colorings, where a Gallai‐coloring is an edge‐coloring of without rainbow triangles. Combining the two concepts, we consider the function that is the minimum number of colors needed for a Gallai‐‐coloring of . Using the anti‐Ramsey number for , we have that is nontrivial only for . We give a general lower bound for this function and we study how this function falls off from being equal to when and to being when . In particular, for appropriate and , we prove that when and , is at most a fractional power of when , and is logarithmic in when .