## 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 (Formula presented.), while Decider wins with high probability if (Formula presented.). This is a factor of (Formula presented.) larger than the lower bound coming from the off-diagonal Ramsey number r(3,s).

Original language | English |
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Pages (from-to) | 1157-1173 |

Number of pages | 17 |

Journal | Random Structures and Algorithms |

Volume | 57 |

Issue number | 4 |

DOIs | |

State | Published - 1 Dec 2020 |

Externally published | Yes |

## Keywords

- Ramsey numbers
- combinatorial games