Authors: Andrzej Grzesik ¹; Mirjana Mikalački ²; Zoltán Lóránt Nagy ³; Alon Naor ⁴; Balázs Patkós ^5,3; Fiona Skerman ⁶

• 1 Theoretical Computer Science Department [Krakow]
• 2 Department of Mathematics and Informatics [Novi Sad]
• 3 Alfréd Rényi Institute of Mathematics
• 4 School of Mathematical Sciences [Tel Aviv]
• 5 Geometric and Algebraic Combinatorics Research Group
• 6 Department of Statistics [Oxford]

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≥3 we analyse the k-star game, where Avoider tries to avoid claiming k edges incident to the same vertex. We consider 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 fmonF, f-F and f+F, where F is the hypergraph of the game.