Abstract
Fix positive integers p and q with 2 ⩽ q ⩽ ( p 2 ) . An edge coloring of the complete graph K n is said to be a ( p , q ) -coloring if every K p receives at least q different colors. The function f ( n , p , q ) is the minimum number of colors that are needed for K n to have a ( p , q ) -coloring. This function was introduced about 40 years ago, but Erdős and Gyárfás were the first to study the function in a systematic way. They proved that f ( n , p , p ) is polynomial in n and asked to determine the maximum q, depending on p, for which f ( n , p , q ) is subpolynomial in n. We prove that the answer is p − 1 .
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