Induced rational exponents and bipartite subgraphs in $K_{s, s}$-free graphs

TL;DR

This paper explores how extremal properties of bipartite graphs can be transferred to induced subgraphs in $K_{s,s}$-free graphs, achieving new results on Turán exponents and induced subgraph structures.

Contribution

It introduces a method to realize rational Turán exponents via smaller forbidden induced bipartite subgraphs and provides supersaturation results for induced trees and cycles in $K_{s,s}$-free graphs.

Findings

Achieved all rational Turán exponents in (1,2) with a small family of induced forbidden bipartite graphs.

Proved supersaturation results for induced trees and cycles in $K_{s,s}$-free graphs.

Provided bounds for $K_{s,s}$-free graphs avoiding certain induced subgraphs, supporting recent conjectures.

Abstract

In this paper, we study a general phenomenon that many extremal results for bipartite graphs can be transferred to the induced setting when the host graph is $K_{s, s}$-free. As manifestations of this phenomenon, we prove that every rational $\frac{a}{b} \in (1, 2), \, a, b \in \mathbb{N}_+$, can be achieved as Tur\'{a}n exponent of a family of at most $2^a$ induced forbidden bipartite graphs, extending a result of Bukh and Conlon [JEMS 2018]. Our forbidden family is a subfamily of theirs which is substantially smaller. A key ingredient, which is yet another instance of this phenomenon, is supersaturation results for induced trees and cycles in $K_{s, s}$-free graphs. We also provide new evidence to a recent conjecture of Hunter, Milojevi\'{c}, Sudakov, and Tomon [JCTB 2025] by proving optimal bounds for the maximum size of $K_{s, s}$-free graphs without an induced copy of theta graphs or prism graphs, whose Tur\'{a}n exponents were determined by Conlon [BLMS 2019] and by Gao, Janzer, Liu, and Xu [IJM 2025+].