Induced rational exponents and bipartite subgraphs in $K_{s, s}$-free graphs
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| Format: | Preprint |
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2025
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| author | Dong, Zichao Gao, Jun Li, Ruonan Liu, Hong |
| author_facet | Dong, Zichao Gao, Jun Li, Ruonan Liu, Hong |
| contents | 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 Turán 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, Milojević, 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 Turán exponents were determined by Conlon [BLMS 2019] and by Gao, Janzer, Liu, and Xu [IJM 2025+]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_09020 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Induced rational exponents and bipartite subgraphs in $K_{s, s}$-free graphs Dong, Zichao Gao, Jun Li, Ruonan Liu, Hong Combinatorics 05C35 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 Turán 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, Milojević, 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 Turán exponents were determined by Conlon [BLMS 2019] and by Gao, Janzer, Liu, and Xu [IJM 2025+]. |
| title | Induced rational exponents and bipartite subgraphs in $K_{s, s}$-free graphs |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2506.09020 |