Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On an Erdős-Szekeres Game

Lara Pudwell

Abstract

We consider a 2-player permutation game inspired by the celebrated Erdős-Szekeres Theorem. The game depends on two positive integer parameters $a$ and $b$ and we determine the winner and give a winning strategy when $a \geq b$ and $b \in \left\{2,3,4,5\right\}$.

On an Erdős-Szekeres Game

Abstract

We consider a 2-player permutation game inspired by the celebrated Erdős-Szekeres Theorem. The game depends on two positive integer parameters and and we determine the winner and give a winning strategy when and .

Paper Structure

This paper contains 11 sections, 8 theorems, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Representation of Moves
  3. Strategy when b=2
  4. Strategy when b=3
  5. Strategy when b=4
  6. Strategy when b=5
  7. Other Directions
  8. Strategy for normal play
  9. Permutations and boards
  10. Number of moves
  11. Number of winning positions

Key Result

Theorem 1

Any permutation of length $n \geq (a-1)(b-1)+1$ contains either an increasing subsequence of length $a$ or a decreasing subsequence of length $b$.

Figures (3)

  • Figure 1: The board corresponding to $\pi=163425$ in a $(6,5)$-game
  • Figure 2: Ordered pairs representing possible next moves in a $(6,5)$-game where $\pi=163425$.
  • Figure 3: A class $\mathcal{P}$ position for the $(a,3)$-game.

Theorems & Definitions (16)

  • Theorem 1
  • proof : Proof of Theorem \ref{['T:ES']}
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Lemma 1
  • ...and 6 more