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Fast winning strategies in a generalized van der Waerden game

Hannah Alpert, Liam Barham, Brian Freidin, Ian Tan, Alexandra Weiner

TL;DR

The paper studies fast winning strategies in a generalized van der Waerden Maker–Breaker game on $\mathbb{N}$, where Maker wins by claiming a homothetic copy $aS+b$ of a finite set $S$. It combines a Hales–Jewett–based upper bound with constructive tree arguments to obtain optimal move counts for small $|S|$, and symmetry-based arguments to rule out fast wins for larger $|S|$. The main contributions are the exact move counts $C_n=n$ for $n\le 3$, $C_4=5$ with a complete characterization of $4$-move-$S$ sets, and a complete resolution showing no $|S|$-move strategy when $|S|\ge 5$, plus explicit tree constructions for size four. These results sharpen the understanding of how quickly Maker can force an affine-copy completion in this affine-geometry flavored combinatorial game, and illustrate the role of symmetry and constructive trees in designing winning strategies.

Abstract

Consider the following Maker-Breaker game. Fix a finite subset $S\subset\mathbb{N}$ of the naturals. The players Maker and Breaker take turns choosing previously unclaimed natural numbers. Maker wins by eventually building a homothetic copy $aS+b$ of $S$, where $a\in\mathbb{N}\setminus\{0\}$ and $b\in\mathbb{Z}$. This is a generalization of the van der Waerden game analyzed by Beck. By the Hales-Jewett theorem, there exists a constant $c$ depending only on $|S|$ such that Maker can win in $c$ or less moves. We show that Maker can win in $|S|$ moves if $|S|\leq 3$. When $|S|=4$, we show that Maker can always win in $5$ or less moves and describe all $S$ such that Maker can win in $4$ moves. If $|S|\geq 5$, Maker has no winning strategy in $|S|$ moves.

Fast winning strategies in a generalized van der Waerden game

TL;DR

The paper studies fast winning strategies in a generalized van der Waerden Maker–Breaker game on , where Maker wins by claiming a homothetic copy of a finite set . It combines a Hales–Jewett–based upper bound with constructive tree arguments to obtain optimal move counts for small , and symmetry-based arguments to rule out fast wins for larger . The main contributions are the exact move counts for , with a complete characterization of -move- sets, and a complete resolution showing no -move strategy when , plus explicit tree constructions for size four. These results sharpen the understanding of how quickly Maker can force an affine-copy completion in this affine-geometry flavored combinatorial game, and illustrate the role of symmetry and constructive trees in designing winning strategies.

Abstract

Consider the following Maker-Breaker game. Fix a finite subset of the naturals. The players Maker and Breaker take turns choosing previously unclaimed natural numbers. Maker wins by eventually building a homothetic copy of , where and . This is a generalization of the van der Waerden game analyzed by Beck. By the Hales-Jewett theorem, there exists a constant depending only on such that Maker can win in or less moves. We show that Maker can win in moves if . When , we show that Maker can always win in or less moves and describe all such that Maker can win in moves. If , Maker has no winning strategy in moves.
Paper Structure (7 sections, 14 theorems, 28 equations, 4 figures, 2 tables)

This paper contains 7 sections, 14 theorems, 28 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Overview of the Paper
  4. Upper Bound for Sets of Fixed Size
  5. Winning Strategies from Trees
  6. A Weak Lower Bound
  7. Sets of Size Four

Key Result

Theorem 1

For all $n,r$ there exists $m$ such that every $r$-coloring $\Delta:[m]\to [r]$ of $[m]$ contains a monochromatic arithmetic progression $S$ of size $n$. That is, $S=\{s,s+a,s+2a,\dots,s+(n-1)a\}\subset [m]$ such that $|\Delta(S)|=1$.

Figures (4)

  • Figure 1: A tree giving a winning strategy for $\{1,2,3,4\}$.
  • Figure 2: A tree giving a winning strategy for $\{0,2,3,6\}$.
  • Figure 3: A tree giving a winning strategy for sets of size 3.
  • Figure 4: A tree giving a winning strategy for symmetric sets of type $(1,4)$.

Theorems & Definitions (32)

  • Theorem 1: Van der Waerden's theorem, Waerden27
  • Definition 1
  • Theorem 2: The Hales-Jewett Theorem, GRS90
  • Theorem 3
  • proof
  • Proposition 1
  • proof
  • Example 1
  • Example 2
  • Corollary 1
  • ...and 22 more