Multicolor Erdős--Rogers Functions

In this paper, we study a multicolor variant of Erdős--Rogers functions. Let f_αₛ; Ki₁, ⋯, Kiₜ(n) be the largest integer m such that there is always an induced Kₛ-free subgraph of size m in every n-vertex graph with a t-edge-coloring in which the edges with the j-th color induce no copy of Kiⱼ. We establish both upper and lower bounds for this multicolor version. Specifically, we show that fα₅; K₃, K₃(n) = n1/2+o(1), Ω(n5/11) ≤ fα₅; K₃, K₃, K₃(n) ≤ n1/2+o(1), and Ω(n20/61) ≤ fα₅; K₃, K₃, K₃, K₃(n) ≤ n1/3+o(1).