Abstract
We study biased $(a:f)$ Avoider-Forcer games played on hypergraphs, in the strict and monotone versions. We give a sufficient condition for Avoider's win, useful in the case of games on hypergraphs whose rank is smaller than $f$. We apply this result to estimate the threshold bias in Avoider-Forcer $(1:f)$ games in which Avoider is trying not to build a copy of a fixed graph $G$ in $K_n$. We also study the $d$-degree $(1:f)$ game in which Avoider's aim is to avoid a spanning subgraph of the minimal degree at least $d$ in $K_n$. We show that the strict 1-degree game has the threshold which is the same as the threshold of the Avoider-Forcer connectivity game.
Highlights
Let V be a finite set and E be a multiset of subsets of V
We study biased (a : f ) Avoider-Forcer games played on hypergraphs, in the strict and monotone versions
We show that the strict 1-degree game has the threshold which is the same as the threshold of the Avoider-Forcer connectivity game
Summary
Let V be a finite set and E be a multiset of subsets of V. Given a hypergraph H, we denote by the lower threshold bias fH− the largest integer such that for every f fH− Forcer has a winning strategy for AF (H, 1, f ). They found a condition sufficient for forcing Avoider to build a copy of a fixed graph G, provided a and f are fixed and n is large From their results we can infer that if f + 1 is coprime to every number not greater than e(G) and f < cn2/e(G) with some constant c depending on G, Avoider in AF (HG,n, 1, f ) is forced to build a copy of G in Kn. Here we will be primarily concerned with the upper and the lower thresholds in d-degree and small-graph games played on Kn. For 1-degree strict game, the result (2).
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