# Protecting billiard balls from collisions

## TL;DR

This paper introduces a game modeling billiard ball collisions, analyzing strategies to maximize or minimize collisions, with bounds related to the Erdős–Szekeres Theorem, revealing new insights into collision configurations.

## Contribution

It formulates a novel game-theoretic problem on billiard collisions and establishes bounds using combinatorial geometry, extending Erdős–Szekeres Theorem applications.

## Key findings

- The game's value is approximately  (up to constants).
- Lower bounds are derived from the Erd
51s-Szekeres Theorem.
- Upper bounds generalize the Erd
51s-Szekeres Theorem.

## Abstract

We present a game inspired by research on the possible number of billiard ball collisions in the whole Euclidean space. One player tries to place $n$ static "balls" with zero radius (i.e., points) in a way that will minimize the total number of possible collisions caused by the cue ball. The other player tries to find initial conditions for the cue ball to maximize the number of collisions. The value of the game is $\sqrt{n}$ (up to constants). The lower bound is based on the Erd\H{o}s-Szekeres Theorem. The upper bound may be considered a generalization of the Erd\H{o}s-Szekeres Theorem.