Maximizing $K_r + I_r$ in graphs with fixed edge density

Abstract

For every integer $r\ge4$, and $\rho \in [0,1]$, we asymptotically determine the maximum proportion of $r$-element sets of vertices that induce either a clique or an independent set in a large graph with density $\rho$. This generalizes a result of Olpp for $r=3$. After the initial idea for the main proof was found by the authors, various AI models were used to streamline the argument and perform the calculations necessary for completion of the proof.