Picker-Chooser fixed graph games

TL;DR

This paper introduces a biased graph game where players aim to control the number of subgraphs isomorphic to a fixed graph H, conjecturing its value aligns with the expected count in a random graph, and proves this for certain H.

Contribution

The paper formulates a new Picker-Chooser fixed graph game and proves the conjecture for specific graph types like complete graphs and trees.

Findings

Conjecture that game value matches expected subgraph count in G(n,p)

Proved the conjecture for complete graphs

Proved the conjecture for trees

Abstract

Given a fixed graph $H$ and a positive integer $n$, a Picker-Chooser $H$-game is a biased game played on the edge set of $K_n$ in which Picker is trying to force many copies of $H$ and Chooser is trying to prevent him from doing so. In this paper we conjecture that the value of the game is roughly the same as the expected number of copies of $H$ in the random graph $G(n,p)$ and prove our conjecture for special cases of $H$ such as complete graphs and trees.