1. Ramsey, Paper, Scissors
- Author
-
Jacob Fox, Xiaoyu He, and Yuval Wigderson
- Subjects
Computer Science::Computer Science and Game Theory ,Applied Mathematics ,General Mathematics ,Combinatorial game theory ,0102 computer and information sciences ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Upper and lower bounds ,Combinatorics ,010201 computation theory & mathematics ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,FOS: Mathematics ,Mathematics - Combinatorics ,Graph (abstract data type) ,Combinatorics (math.CO) ,Ramsey's theorem ,Null graph ,Software ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics ,Independence number - 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 $0B\sqrt{n}\log{n}$. This is a factor of $\Theta(\sqrt{\log{n}})$ larger than the lower bound coming from the off-diagonal Ramsey number $r(3,s)$.
- Published
- 2020