Creating triangles in Constructor-Blocker games

TL;DR

The paper investigates a two-player Constructor-Blocker game that models generalized Turán problems by asking how many copies of a fixed subgraph $H$ (here triangles) can appear in Constructor's $F$-free graph under optimal play. It develops and applies tools from extremal graph theory and scoring positional games, including adaptations of the Erdős–Selfridge criterion, to obtain precise asymptotics and bounds across multiple forbidden-subgraph scenarios and planarity constraints. Key findings include $g(n,K_3,∅) = (1+o(1)) n^3/48$, $g(n,K_3,P_6) = (1+o(1)) n/2$, $g(n,K_3,S_4) = (1+o(1)) n/3$, and bounds for $g(n,K_3,S_5)$; under planar constraints, $g_{PCB}(n,K_3) = (1+o(1)) 3n$ while ECB yields $(1+o(1)) n/2 ≤ g_{ECB}(n,K_3) ≤ (1+o(1)) 2n/3$. These results deepen the connection between combinatorial game theory and extremal graph theory, offering exact asymptotics and guiding directions for related Maker-Breaker and Avoider-Enforcer frameworks under structural constraints.

Abstract

Generalized Turán problems investigate the maximization of the number of certain structures (typically edges) under some constraints in a graph. We study a game version of these problems, the Constructor-Blocker game. We mainly focus on the case where Constructor tries to maximize the number of triangles in her graph, while forbidding her to claim short paths or cycles. We also study a variant of this game, where we impose some planarity constraints on Constructor instead of forbidding certain subgraphs. For all games studied, we obtain (precise) asymptotics or upper and lower bounds.

Figures (17)

• Figure 2.1: How to build a triangle with a pendant leg? An example of Constructor's (solid lines) and Blocker's (dotted lines) successive moves, Constructor being the first player.
• Figure 3.1: How connected components with many triangles in a $P_6$-free graph can look like. The curved lines mean that there might or might not be such edges.
• Figure 3.2: Constructor's strategy when forbidding $P_6$
• Figure 3.3: Notations for the proof of Theorem \ref{['thm : g(n,K_3,P_6)']}
• Figure 3.4: Constructor's strategy when $S_4$ is forbidden

Theorems & Definitions (55)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1: Erdős-Selfridge-Criterion ERDOS1973298
• Theorem 2.1: Theorem 10 in bagan_incidence_2024
• Lemma 2.2