Restricted Positional Games
Pranav Avadhanam, Siddhartha G. Jena

TL;DR
This paper introduces a new restricted positional game called Connect-Tac-Toe, generalizes Connect-4 to hypercubes, and analyzes its restrictiveness using a variant of the Hales-Jewett number, providing bounds for it.
Contribution
It defines a novel restricted positional game, Connect-Tac-Toe, and develops a framework to analyze its restrictiveness through a modified Hales-Jewett number.
Findings
Introduced Connect-Tac-Toe as a new restricted positional game.
Established a logarithmic lower bound for the game’s Hales-Jewett number.
Connected the game’s restrictiveness to hypercube structures.
Abstract
A positional game is a game where two players sequentially label vertices of a hypergraph, consisting of a board and a collection of winning sets, with colors assigned to each player until all vertices of the board are claimed. The first player to claim all elements of a winning set wins. If no player claims all the elements of a winning set, then the game results in a draw. One such example of a positional game is Tic-Tac-Toe, where the board is the 3-by-3 grid. The popular game of Connect-4 is an example of what we define to be a "restricted positional game". Here, we introduce another example of a restricted positional game, Connect-Tac-Toe, which additionally has the influence of gravity-like restrictions to affect what plays are possible. It is a generalization of the Connect-4 game for the hypercube. We define a variant of the Hales-Jewett number for this game as a way to classify…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Graph Labeling and Dimension Problems
