A Winning Strategy for the Ramsey Graph Game

Abstract

We consider a 'Maker-Breaker' version of the Ramsey Graph Game, RG(n), and present a winning strategy for Maker requiring at most (« -3)2"~' + n + 1 moves. This is the fastest winning strategy known so far. We also demonstrate how the ideas presented can be used to develop winning strategies for some related combinatorial games.

Fast winning strategies in Maker–Breaker games

“…The best upper bound, f (q) = O ((q − 3)2 q−1 ) is due to Pekeč [12]. Beck proved that the exponential dependency on q cannot be avoided, namely f (q) = Ω( √ 2 q ) (see [3]).…”

Positional Games and the Second Moment Method

“…Two edges determine at least 3 vertices, and n 3 is bigger than O(n 2 ), the duration of a play. By (6)- (7) and (15) we obtain The best lower bound result known before was the "half as good" log 2 n− O(1) < mb(n), see Beck [4] and Pekec [14].…”

“…Now we see that the strongest condition is (14), which by Stirling's formula means (15) q ≤ 2 log 2 n − 2 log 2 log 2 n + 2 log 2 e − 10 3 + o(1).…”

“…Since the Maker's win criterion is identical to the Chooser's win criterion, by repeating the calculations of (8)- (14), we conclude that Chooser wins if (50) q ≤ 2 log 2 n − 2 log 2 log 2 n + 2 log 2 e − 10 3…”

Fast winning strategies in positional games

“…It is known that, playing on the edges of K n , Maker can build a q-clique in a constant (depending on q, but not on n) number of moves, that is, τ (K q n ) = f (q), where the hyperedges of K q n are the q-cliques of K n . The best upper bound, f (q) = O((q −3)2 q−1 ) is due to Pekeč [12]. Beck proved that the exponential dependency on q cannot be avoided, namely f (q) = Ω( √ 2 q ) (see [3]).…”