## Almost-rainbow edge-colorings of some small subgraphs

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##### Authors

Elliot Krop

Irina Krop

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##### Type

Journal Article, Academic Journal

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##### Abstract

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 Erd\H os and Gy\'arf\'as. We show that $f(n,5,9) \geq \frac{7}{4}n-3$, slightly improving the bound of Axenovich. We make small improvements on bounds of Erd\H os and Gy\'arf\'as by showing $\frac{5}{6}n+1\leq f(n,4,5)$ 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.