On the extremal number of incidence graphs

TL;DR

The paper addresses the extremal problem ex$(n,H)$ for bipartite incidence-type graphs derived from polytopes, introducing a general upper bound via a purely algebraic reflection-group framework. By encoding repeated Cauchy–Schwarz applications as cut involutions in finite Coxeter groups, the authors prove that generalised face-incidence graphs satisfy ex$(n,H)=O(n^{2-c})$ with $c=(v(H)-t-1)/(e(H)-t)$, where $t$ is the larger bipartition size. This unifies and extends prior work of Janzer–Sudakov and Conlon–Lee, confirming conjectures for broader incidence-graph families, including certain $K_{r,r}$-free graphs, and yields concrete bounds for examples like the 24-cell. The approach emphasizes percolating sequences and folding operations in the Coxeter framework, offering a versatile algebraic tool for extremal inequalities and norming properties of reflection graphs. Overall, the work broadens the repertoire of graphs for which tight extremal bounds can be derived via percolation-inspired Cauchy–Schwarz analysis.

Abstract

Given a graph $H$ and a natural number $n$, the extremal number $\mathrm{ex}(n, H)$ is the largest number of edges in an $n$-vertex graph containing no copy of $H$. In this paper, we obtain a general upper bound for the extremal number of generalised face-incidence graphs, a family which includes the standard face-incidence graphs of regular polytopes. This builds on and generalises work of Janzer and Sudakov, who obtained the same bound for hypercubes and bipartite Kneser graphs, and allows us to confirm a conjecture of Conlon and Lee on the extremal number of $K_{r,r}$-free bipartite graphs for certain incidence graphs. In their work, Janzer and Sudakov showed that such an upper bound on the extremal number holds whenever the graph $H$ satisfies a certain percolation property which captures an appropriate sequence of repeated applications of the Cauchy--Schwarz inequality, a property which they then verify for hypercubes and bipartite Kneser graphs. This percolation property bears close resemblance to a property that arose in earlier work of Conlon and Lee on weakly norming graphs. In this latter work, Conlon and Lee developed a method for controlling repeated applications of the Cauchy--Schwarz inequality based on the properties of reflection groups, which then allowed them to isolate a broad family of weakly norming graphs. Here, we develop this method further, casting it in a purely algebraic form that allows us not only to combine it with the Janzer--Sudakov result and obtain the desired result about the extremal number of incidence graphs, but also to simplify the proofs of both the Conlon--Lee result on weakly norming graphs and a related result of Coregliano.

Figures (2)

• Figure 1: The $2$-blow-up of $C_6$
• Figure 2: The 24-cell and its $(0,3)$-incidence graph

Theorems & Definitions (40)

• Conjecture 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3: Lemma 1.4.1 in BB05
• Theorem 2.4: Theorem 1.4.3 in BB05
• Proposition 2.5: Proposition 1.4.7 in BB05
• Corollary 2.6: Corollary 1.4.8 in BB05
• Theorem 2.7: Theorem 2.2.2 and Corollary 2.2.3 in BB05
• Theorem 2.8: Theorem 2.2.6 in BB05