A Strategy for Maker in the Clique Game which Helps to Tackle some Open Problems by Beck

TL;DR

This paper introduces new strategies for Maker in clique and tournament games on complete graphs, achieving larger cliques and tournaments than previously possible, and partially solving open problems posed by Beck.

Contribution

It presents novel bounds for Maker's ability to form large cliques and tournaments in Maker/Breaker games, improving previous results and addressing open problems by Beck.

Findings

Maker can achieve a K_q with q proportional to log N in the (m:b) clique game.

Maker can form a K_q in 2^(2q/3) moves in the (1:1) clique game.

Maker wins the tournament game with q approximately log_2(N), matching random graph thresholds.

Abstract

We study Maker/Breaker games on the edges of the complete graph, as introduced by Chvatal and Erdos. We show that in the (m:b) clique game played on K_{N}, the complete graph on N vertices, Maker can achieve a K_{q} for q = (m/(log_{2}(b + 1)) - o(1)) * log N, which partially solves an open problem by Beck. Moreover, we show that in the (1:1) clique game played on K_{N} for a sufficiently large N, Maker can achieve a K_{q} in only 2^(2q/3) moves, which improves the previous best bound and answers a question of Beck. Finally we consider the so called tournament game. A tournament is a directed graph where every pair of vertices is connected by a single directed edge. The tournament game is played on K_{N}. At the beginning Breaker fixes an arbitrary tournament T_{q} on q vertices. Maker and Breaker then alternately take turns at claiming one unclaimed edge e and selecting one of the two possible orientations. Maker wins if his graph contains a copy of the goal tournament T_{q}; otherwise Breaker wins. We show that Maker wins the tournament game on K_{N} with q = (1 - o(1))*log_{2}(N) which supports the random graph intuition: the threshold for q is asymptotically the same for the game played by two "clever'' players and the game played by two ``random'' players. This last result solves an open problem of Beck which he included in his list of the seven most humiliating open problems.