On color isomorphic subdivisions

Abstract

Given a graph $H$ and an integer $k⩾2$, let $f_k(n,H)$ be the smallest number of colors $C$ such that there exists a proper edge-coloring of the complete graph $K_n$ with $C$ colors containing no $k$ vertex-disjoint color isomorphic copies of $H$. In this paper, we prove that $f_2(n,H_t)=\Omega (n^{1+\frac{1}{2t-3}})$ where $H_t$ is the $1$-subdivision of the complete graph $K_t$. This answers a question of Conlon and Tyomkyn (arXiv: 2002.00921).

9 pages