Avoidance Games Are PSPACE-Complete

Avoidance games are games in which two players claim vertices of a hypergraph and try to avoid some structures. These games are studied since the introduction of the game of SIM in 1968, but only few complexity results are known on them. In 2001, Slany proved some partial results on Avoider-Avoider games complexity, and in 2017 Bonnet et al. proved that short Avoider-Enforcer games are Co-W[1]-hard. More recently, in 2022, Miltzow and Stojakovi\'c proved that these games are NP-hard. As these games corresponds to the mis\`ere version of the well-known Maker-Breaker games, introduced in 1963 and proven PSPACE-complete in 1978, one could expect these games to be PSPACE-complete too, but the question remained open since then. We prove here that both Avoider-Avoider and Avoider-Enforcer conventions are PSPACE-complete, and as a consequence of it that some particular Avoider-Enforcer games also are

Avoider-Enforcer star games

In this paper, we study $(1 : b)$ Avoider-Enforcer games played on the edge set of the complete graph on $n$ vertices. For every constant $k\geq 3$ we analyse the $k$-star game, where Avoider tries to avoid claiming $k$ edges incident to the same vertex. We analyse both versions of Avoider-Enforcer games -- the strict and the monotone -- and for each provide explicit winning strategies for both players. We determine the order of magnitude of the threshold biases $f^{mon}_\mathcal{F}$, $f^-_\mathcal{F}$ and $f^+_\mathcal{F}$, where $\mathcal{F}$ is the hypergraph of the game

Fast winning strategies in Avoider-Enforcer games

In numerous positional games the identity of the winner is easily determined. In this case one of the more interesting questions is not {\em who} wins but rather {\em how fast} can one win. These type of problems were studied earlier for Maker-Breaker games; here we initiate their study for unbiased Avoider-Enforcer games played on the edge set of the complete graph $K_n$ on $n$ vertices. For several games that are known to be an Enforcer's win, we estimate quite precisely the minimum number of moves Enforcer has to play in order to win. We consider the non-planarity game, the connectivity game and the non-bipartite game

Keeping Avoider's graph almost acyclic

We consider biased $(1:b)$ Avoider-Enforcer games in the monotone and strict versions. In particular, we show that Avoider can keep his graph being a forest for every but maybe the last round of the game if $b \geq 200 n \ln n$. By this we obtain essentially optimal upper bounds on the threshold biases for the non-planarity game, the non-$k$-colorability game, and the $K_t$-minor game thus addressing a question and improving the results of Hefetz, Krivelevich, Stojakovi\'c, and Szab\'o. Moreover, we give a slight improvement for the lower bound in the non-planarity game.Comment: 11 page

Positional Games

Positional games are a branch of combinatorics, researching a variety of two-player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex, to purely abstract games played on graphs and hypergraphs. It is closely connected to many other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic combinatorics, and to computer science. We survey the basic notions of the field, its approaches and tools, as well as numerous recent advances, standing open problems and promising research directions.Comment: Submitted to Proceedings of the ICM 201