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On the Constructor-Blocker Game

József Balogh, Ce Chen, Sean English

TL;DR

This work investigates the Constructor-Blocker edge-claiming game on the complete graph, focusing on the score $g(n,H,F)$, the maximum number of copies of $H$ achievable by a first-player Constructor who must keep her graph $F$-free. It extends prior results by solving $g(n,H,F)$ when $H=K_r$ and $oldsymbol{oldsymbol{\chi(F)>r}}$, and by analyzing both $H$ and $F$ as odd cycles via Szemerédi's Regularity Lemma; it also provides bounds for the case $H=K_3$ and $F=K_{2,2}$. The proofs combine JumbleG-type pseudorandomness arguments, bipartite variants, and blow-up/construction techniques, augmented by 2-fold Sidon set methods to create favorable hypergraph structures. Overall, the results advance generalized Turán theory in the context of Maker-Breaker style graph games, delivering precise asymptotics in several natural regimes and shedding light on the interplay between extremal graph structure and game dynamics.

Abstract

In the Constructor-Blocker game, two players, Constructor and Blocker, alternatively claim unclaimed edges of the complete graph $K_n$. For given graphs $F$ and $H$, Constructor can only claim edges that leave her graph $F$-free, while Blocker has no restrictions. Constructor's goal is to build as many copies of $H$ as she can, while Blocker attempts to stop this. The game ends once there are no more edges that Constructor can claim. The score $g(n,H,F)$ of the game is the number of copies of $H$ in Constructor's graph at the end of the game, when both players play optimally and Constructor plays first. In this paper, we extend results of Patkós, Stojaković and Vizer on $g(n, H, F)$ to many pairs of $H$ and $F$: We determine $g(n, H, F)$ when $H=K_r$ and $χ(F)>r$, also when both $H$ and $F$ are odd cycles, using Szemerédi's Regularity Lemma. We also obtain bounds of $g(n, H, F)$ when $H=K_3$ and $F=K_{2,2}$.

On the Constructor-Blocker Game

TL;DR

This work investigates the Constructor-Blocker edge-claiming game on the complete graph, focusing on the score , the maximum number of copies of achievable by a first-player Constructor who must keep her graph -free. It extends prior results by solving when and , and by analyzing both and as odd cycles via Szemerédi's Regularity Lemma; it also provides bounds for the case and . The proofs combine JumbleG-type pseudorandomness arguments, bipartite variants, and blow-up/construction techniques, augmented by 2-fold Sidon set methods to create favorable hypergraph structures. Overall, the results advance generalized Turán theory in the context of Maker-Breaker style graph games, delivering precise asymptotics in several natural regimes and shedding light on the interplay between extremal graph structure and game dynamics.

Abstract

In the Constructor-Blocker game, two players, Constructor and Blocker, alternatively claim unclaimed edges of the complete graph . For given graphs and , Constructor can only claim edges that leave her graph -free, while Blocker has no restrictions. Constructor's goal is to build as many copies of as she can, while Blocker attempts to stop this. The game ends once there are no more edges that Constructor can claim. The score of the game is the number of copies of in Constructor's graph at the end of the game, when both players play optimally and Constructor plays first. In this paper, we extend results of Patkós, Stojaković and Vizer on to many pairs of and : We determine when and , also when both and are odd cycles, using Szemerédi's Regularity Lemma. We also obtain bounds of when and .
Paper Structure (12 sections, 16 theorems, 37 equations)

This paper contains 12 sections, 16 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results of generalized Turán number
  4. The game of JumbleG and Szemerédi's Regularity Lemma
  5. 2-fold Sidon sets
  6. Proof of Theorem \ref{['clique-other']}
  7. Proofs of Theorems \ref{['k3 and odd']} and \ref{['odd-odd']}
  8. 2 for 2
  9. 2 for 2
  10. 2 for 2
  11. Proof of Theorem \ref{['k3-k22']}
  12. Acknowledgment

Key Result

Theorem 2

If $\chi(F)=s\geq 3$ and $n$ is large enough, then Additionally, the last inequality also holds when $s=2$.

Theorems & Definitions (26)

  • Theorem 2: Theorem 2 in BM
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Theorem 6
  • Theorem 7
  • Conjecture 8
  • Theorem 9: Zykov zykov, Erdős ErdosCliques
  • Theorem 10: Proposition 2.1 in AS
  • Theorem 11: Proposition 2.2 in AS
  • ...and 16 more