New Bounds for Induced Turán Problems

TL;DR

The paper investigates the induced extremal number $\mathrm{ex}(n,H\text{-ind})$ in graphs forbidding $K_{s,s}$ and induced copies of a bipartite pattern $H$, aiming to relate it to the standard $\mathrm{ex}(n,H)$. It develops a framework of reductions to connected components and to the $2$-core, and proves a degeneracy-based upper bound $\mathrm{ex}(n,{K_{s,s}, H\text{-ind}}) \le O(n^{2-1/(20 r^4)})$ for $r$-degenerate bipartite $H$ using dependent random choice, with refined bounds when forbidding specific non-edges of $H$. The authors also establish bounds under edge-forbidding scenarios (Cases 1–2) that improve ex-dependence on degeneracy, and discuss potential incidence-geometry counterexamples that could challenge the conjecture relating induced and standard extremal numbers. Overall, the work connects induced extremal behavior to the degeneracy structure of $H$ and motivates further exploration of possible counterexamples from geometric incidence graphs. These results advance understanding of how induced-avoidance interacts with classical Turán-type extremal theory and identify key directions for tightening bounds and assessing conjectures.

Abstract

In a recent paper, Hunter, Milojević, Sudakov and Tomon consider the maximum number of edges in an $n$-vertex graph containing no copy of the complete bipartite graph $K_{s,s}$ and no induced copy of a "pattern" graph $H$. They conjecture that, for $s \geq |V(H)|$, this "induced extremal number" differs by at most a constant factor from the standard extremal number of $H$. Towards this, we give bounds on the induced extremal number in terms of degeneracy, which establish some non-trivial relationship between the induced and standard extremal numbers in general. We also show that (as in the case of standard extremal numbers) the induced extremal number is dominated by that of the 2-core of a single connected component. Finally, we present some graphs arising from incidence geometry which may serve as counterexamples to the conjecture.

Figures (3)

• Figure 1: $\mathop{\mathrm{Hea}}\nolimits^-$, the incidence graph of the Fano plane with a single edge deleted. Deleted edge shown dashed.
• Figure 2: A complete quadrangle, with a line between two of the three diagonal points. The dashed extension of that line indicates that it passes through the third diagonal if and only if the underlying field has characteristic $2$ (in which case the configuration is isomorphic to the Fano plane).
• Figure 3: Two graphs $H$ where the bound $\mathop{\mathrm{ex}}\nolimits(n, \{K_{2,2}, H\text{-ind}\}) = \Theta(n^{3/2})$ can be obtained from incidence geometry theorems: the Pappus graph (left), and the Desargues graph (right), each with a single deleted edge (denoted by a dashed line). Pappus's theorem and Desargues's theorem, respectively, ensure that neither can appear as an induced subgraph of $\mathop{\mathrm{PG}}\nolimits(2, q)$ for any $q$, since the dashed edge will always be present.

Theorems & Definitions (47)

• conjecture 1.1: main-paper
• proposition 1.1
• definition 1.2
• theorem 1.2
• definition 1.3
• theorem 1.3
• example 2.1
• conjecture 2.2
• proposition 2.2
• proof