Game saturation of intersecting families

TL;DR

This paper studies a combinatorial game involving two players claiming intersecting subsets of a finite set, analyzing the saturation number under optimal play and establishing bounds based on parameters n and k.

Contribution

The paper introduces and analyzes a new combinatorial game, providing bounds on the saturation number for both starting players under optimal strategies.

Findings

Established lower bound: m  upper bound: O_k(n^{k-\u221a{k}/2})

Derived bounds depend on parameters n and k, showing how game complexity scales

Analyzed game saturation for both first and second players under optimal strategies

Abstract

We consider the following combinatorial game: two players, Fast and Slow, claim $k$-element subsets of $[n]=\{1,2,...,n\}$ alternately, one at each turn, such that both players are allowed to pick sets that intersect all previously claimed subsets. The game ends when there does not exist any unclaimed $k$-subset that meets all already claimed sets. The score of the game is the number of sets claimed by the two players, the aim of Fast is to keep the score as low as possible, while the aim of Slow is to postpone the game's end as long as possible. The game saturation number is the score of the game when both players play according to an optimal strategy. To be precise we have to distinguish two cases depending on which player takes the first move. Let $gsat_F(\mathbb{I}_{n,k})$ and $gsat_S(\mathbb{I}_{n,k})$ denote the score of the saturation game when both players play according to an optimal strategy and the game starts with Fast's or Slow's move, respectively. We prove that $\Omega_k(n^{k/3-5}) \le gsat_F(\mathbb{I}_{n,k}),gsat_S(\mathbb{I}_{n,k}) \le O_k(n^{k-\sqrt{k}/2})$ holds.