On a clique-building game of Erd\H{o}s

Abstract

The following game was introduced in a list of open problems from 1983 attributed to Erd\H{o}s: two players take turns claiming edges of a $K_n$ until all edges are exhausted. Player 1 wins the game if the largest clique that they claim at the end is strictly larger than the largest clique of their opponent; otherwise, Player 2 wins the game. Erd\H{o}s conjectured that Player 2 always wins this game for $n\geq 3$. We make the first known progress on this problem, proving that this holds for at least $3/4$ of all such $n$. We also address a biased version of this game, as well as the corresponding degree-building game, both of which were originally proposed by Erd\H{o}s as well.