On a conjecture of erdöus, simonovits, and sós concerning anti‐Ramsey theorems

For n ≧ k ≧ 3, let f(n,C^k) denote the maximum number m for which it is possible to color the edges of the complete graph Kⁿ with m colors in such a way that each k‐cycle C^k in Kⁿ has atleast two edges of the same color. Erdös, Simonovits, and Sós conjectured that for every fixed k ≧ 3, f(n, C^k) = n (k–2/2 + 1/k–1) + O(1), and proved it only for k = 3. It is shown that f(n, C⁴) = n + [1/3n] – 1, and the conjecture thus proved for k = 4. Some upper bounds are also obtained for f(n, C^k), k ≧ 5.