On the Erdos-Simonovits-Sos conjecture about the anti-Ramsey number of a cycle

Given a positive integer n and a family F of graphs, let f(n, F) denote the maximum number of colours in an edge-colouring of K,, such that no subgraph of K,, belonging to F has distinct colours on its edges. Erdos, Simonovits and Sos [6] conjectured for fixed k with k greater than or equal to 3 that f(n,C-k) = (k-2/2)n + O(1). This has been proved for k less than or equal to 7. For general k, in this paper we improve the previous bound of (k - 2)n - ( k 2 1) to f(n, C-k) less than or equal to (k+1/2 - 2/k-1) n - (k - 2). For even k, we further improve it to k/2 n - (k - 2). We also prove that f(n,{C-k, Ck+1, Ck+2}) less than or equal to (k-2/2 + 1/k-1) n - 1, which is sharp.