# Ramsey, Paper, Scissors

## TL;DR

This paper introduces a new graph Ramsey game involving two players, analyzing the threshold at which Proposer can ensure a large independent set, revealing a phase transition in winning probabilities.

## Contribution

It establishes the existence of a threshold phenomenon in the Ramsey, Paper, Scissors game with optimal randomized strategies, extending understanding of probabilistic combinatorial games.

## Key findings

- Proposer wins with high probability if s < A√n log n
- Decider wins with high probability if s > B√n log n
- Threshold factor is larger than classical Ramsey bounds by Θ(√log n)

## Abstract

We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph on $n$ vertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when Proposer has no legal moves remaining, and Proposer wins if the final graph has independence number at least $s$. We prove a threshold phenomenon exists for this game by exhibiting randomized strategies for both players that are optimal up to constants. Namely, there exist constants $0<A<B$ such that (under optimal play) Proposer wins with high probability if $s<A\sqrt{n}\log{n}$, while Decider wins with high probability if $s>B\sqrt{n}\log{n}$. This is a factor of $\Theta(\sqrt{\log{n}})$ larger than the lower bound coming from the off-diagonal Ramsey number $r(3,s)$.