Maker-breaker percolation games II: Escaping to infinity

Day, A. Nicholas; Falgas–Ravry, Victor

doi:10.1016/j.jctb.2020.06.006

Cited by 3 publications

(4 citation statements)

References 17 publications

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“…Note that the condition n q + 1 is necessary, as if n q, then V could win the game in a single turn. We in fact prove a more general result, showing H can win the q-double-response game (defined below); this will not complicate the argument, and the greater generality will allow us to apply these results to the study of percolation games in the sequel to this paper [11]. A key idea in our proof will be to consider a third game, the secure game, where V plays one edge at at time but is given the extra power of reclaiming some of H 's edges.…”

Section: The (2q Q)-crossing Game: Maker Wins On Arbitrarily Long Anmentioning

confidence: 80%

“…Unsurprisingly, crossing games turned out to play an important role in our arguments when studying percolation games. In particular, the results we establish in this paper are key ingredients in the proofs of our main results on Maker-Breaker percolation games that we establish in the sequel [11] to the present paper.…”

Section: Background and Motivationmentioning

confidence: 82%

“…A different strategy can be read out of a winning strategy due to Lehman [19] for a different combinatorial game, known as the Shannon switching game. In addition to the crossing games studied in this paper, ideas related to Lehman's winning strategy for the Shannon switching game play important role in our study of Maker-Breaker percolation games in our companion paper [11]. For these reasons and for completeness, we describe the Shannon switching game and its application to Bridg-it in detail below.…”

Section: The (1 1)-crossing Game: Bridg-it and The Shannon Switchingmentioning

confidence: 99%

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Maker–Breaker percolation games I: crossing grids

Day

Falgas–Ravry

2020

Combinator. Probab. Comp.

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Motivated by problems in percolation theory, we study the following two-player positional game. Let Λm×n be a rectangular grid-graph with m vertices in each row and n vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims p (as yet unclaimed) edges of the board Λm×n, while on each of his turns Breaker claims q (as yet unclaimed) edges of the board and destroys them. Maker wins the game if she manages to claim all the edges of a crossing path joining the left-hand side of the board to its right-hand side, otherwise Breaker wins. We call this game the (p, q)-crossing game on Λm×n. Given m, n ∈ ℕ, for which pairs (p, q) does Maker have a winning strategy for the (p, q)-crossing game on Λm×n? The (1, 1)-case corresponds exactly to the popular game of Bridg-it, which is well understood due to it being a special case of the older Shannon switching game. In this paper we study the general (p, q)-case. Our main result is to establish the following transition. If p ≥ 2q, then Maker wins the game on arbitrarily long versions of the narrowest board possible, that is, Maker has a winning strategy for the (2q, q)-crossing game on Λm×(q+1) for any m ∈ ℕ. If p ≤ 2q − 1, then for every width n of the board, Breaker has a winning strategy for the (p, q)-crossing game on Λm×n for all sufficiently large board-lengths m. Our winning strategies in both cases adapt more generally to other grids and crossing games. In addition we pose many new questions and problems.

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Section: The (2q Q)-crossing Game: Maker Wins On Arbitrarily Long Anmentioning

confidence: 80%

Section: Background and Motivationmentioning

confidence: 82%

Section: The (1 1)-crossing Game: Bridg-it and The Shannon Switchingmentioning

confidence: 99%

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Maker–Breaker percolation games I: crossing grids

Day

Falgas–Ravry

2020

Combinator. Probab. Comp.

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“…For recent results about Maker–Breaker games on infinite graphs we refer to [3, 12]. Some games (like the base-exchange game in [1]) can be phrased more naturally in the language of infinite matroids.…”

Section: Introductionmentioning

confidence: 99%

Maker–breaker Games on And

BOWLER¹,

GUT²,

Joó³

et al. 2022

J. symb. log.

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We investigate Maker–Breaker games on graphs of size $\aleph _1$ in which Maker’s goal is to build a copy of the host graph. We establish a firm dependence of the outcome of the game on the axiomatic framework. Relating to this, we prove that there is a winning strategy for Maker in the $K_{\omega ,\omega _1}$ -game under ZFC+MA+ $\neg $ CH and a winning strategy for Breaker under ZFC+CH. We prove a similar result for the $K_{\omega _1}$ -game. Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under ZFC+CH again.

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