On a clique-building game of Erdős

Abstract

The following game was introduced in a list of open problems from 1983 attributed to Erdős: 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. Erdős conjectured that Player 2 always wins this game for $n\ge 3$. 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 Erdős as well.

13 pages; accepted at Journal of Graph Theory