On a problem of C. C. Chen and D. E. Daykin

AbstractLet α(k, p, h) be the maximum number of vertices a complete edge-colored graph may have with no color appearing more than k times at any vertex and not containing a complete subgraph on p vertices with no color appearing more than h times at any vertex. We prove that α(k, p, h) ≤ h + 1 + (k − 1){(p − h − 1) × (hp + 1)}1h and obtain a stronger upper bound for α(k, 3, 1). Further, we prove that a complete edge-colored graph with n vertices contains a complete subgraph on p vertices in which no two edges have the same color if (n3)>(p3)Σi=1t(ei2) where ei is the number of edges of color i, 1 ≤ i ≤ t