Almost-rainbow edge-colorings of some small subgraphs

Let $f(n,p,q)$ be the minimum number of colors necessary to color the edges of $K_n$ so that every $K_p$ is at least $q$-colored. We improve current bounds on the {7/4}n-3$, slightly improving the bound of Axenovich. We make small improvements on bounds of Erd\H os and Gy\'arf\'as by showing${5/6}n+1\leq f(n,4,5)$and for all even$n\not\equiv 1 \pmod 3$,$f(n,4,5)\leq n-1$. For a complete bipartite graph$G=K_{n,n}$, we show an n-color construction to color the edges of$G$so that every$C_4\subseteq G$ is colored by at least three colors. This improves the best known upper bound of M. Axenovich, Z. F\"uredi, and D. Mubayi.Comment: 13 page

Almost-Rainbow Edge-Colorings of Some Small Subgraphs

Let $f(n, p, q)$ be the minimum number of colors necessary to color the edges of $K_n$ so that every $K_p$ is at least $q$-colored. We improve current bounds on these nearly “anti-Ramsey” numbers, first studied by Erdös and Gyárfás. We show that $f(n, 5, 0) \ge \frac{7}{4} n - 3$, slightly improving the bound of Axenovich. We make small improvements on bounds of Erdös and Gyárfás by showing $\frac{5}{6} n + 1 \leq f(n,4,5)$ and for all even $n ≢ 1(\text{mod } 3)$, $f(n, 4, 5) \leq n−1$. For a complete bipartite graph $G= K_{n,n}$, we show an $n$-color construction to color the edges of $G$ so that every $C_4 ⊆ G$ is colored by at least three colors. This improves the best known upper bound of Axenovich, Füredi, and Mubayi