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On a clique-building game of Erdős

Alexandru Malekshahian, Sam Spiro

Abstract

The following game was introduced in a list of open problems from 1983 attributed to Erdős: two players take turns claiming edges of a $K_n$ until all edges are exhausted. Player 1 wins the game if the largest clique that they claim at the end is strictly larger than the largest clique of their opponent; otherwise, Player 2 wins the game. Erdős conjectured that Player 2 always wins this game for $n\geq 3$. We make the first known progress on this problem, proving that this holds for at least $3/4$ of all such $n$. We also address a biased version of this game, as well as the corresponding degree-building game, both of which were originally proposed by Erdős as well.

On a clique-building game of Erdős

Abstract

The following game was introduced in a list of open problems from 1983 attributed to Erdős: two players take turns claiming edges of a until all edges are exhausted. Player 1 wins the game if the largest clique that they claim at the end is strictly larger than the largest clique of their opponent; otherwise, Player 2 wins the game. Erdős conjectured that Player 2 always wins this game for . We make the first known progress on this problem, proving that this holds for at least of all such . We also address a biased version of this game, as well as the corresponding degree-building game, both of which were originally proposed by Erdős as well.

Paper Structure

This paper contains 5 sections, 10 theorems, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The clique-building game
  3. The star-building game
  4. The biased clique-building game
  5. Concluding remarks

Key Result

Theorem 1

Let $\mathcal{C}\subseteq \mathbb{N}$ be the set of all natural numbers $n$ for which the Clique$(n)$ game in a Player 2 win. Then $\mathcal{C}$ has asymptotic density at least $\frac{3}{4}$.

Figures (4)

  • Figure 1: The game state before Blue claims his first edge. We are guaranteed a strictly larger Blue clique than Red inside the $K_n$ subgraph indicated above when Blue follows Player 1's winning strategy from $\mathrm{Clique}(n)$.
  • Figure 2: The position $S_1$.
  • Figure 3: The five possible game states after Red's second move given that Blue played his first move incident to Red's first move.
  • Figure 4: The position $S_2(H)$.

Theorems & Definitions (30)

  • Definition
  • Conjecture : Erdős guy83
  • Theorem 1
  • Definition
  • Theorem 2
  • Definition
  • Theorem 3
  • Remark 4
  • Proposition 5
  • proof
  • ...and 20 more