On color isomorphic subdivisions (Q2113360)

For \(k \geq 2\) and graph \(H\), let \(f_k(n,H)\) denote the smallest positive integer \(C\) such that there is 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 work, the authors study the growth rate of \(f_2(n,H_t)\), where \(H_t\) is the 1-subdivision of \(K_t\), \(t\geq 3\). Specifically, in the main result of the paper, they prove that \[ f_2(n,H_t) = \Omega \Big(n^{1+\frac{1}{2t-3}}\Big).\] The introduction of the function \(f_k(n,H)\) originated from the work of \textit{D. Conlon} and \textit{M. Tyomkyn} [SIAM J. Discrete Math. 35, No. 3, 2249--2264 (2021; Zbl 1478.05049)].