2024/10/23 by Malekshahian, Alexandru, Spiro, Sam
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2410.18304
The following game was introduced in a list of open problems from 1983 attributed to Erdős: two players take turns claiming edges of a Kn 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≥ 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.