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arXiv · 2608.15777

A finite forbidden family with superlinear surplus and no three-factor product extremizers

Chuandong Xu
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Abstract

We construct a fixed finite family $\mathcal L$ of ordinary forbidden subgraphs with $p(\mathcal L)=3$ and a constant $c>0$ such that $$ex(n,\mathcal L)>t_3(n)+cn^{3/2}$$ at every sufficiently large order. Nevertheless, the complement of every sufficiently large $\mathcal L$-extremal graph has at most two connected components. In particular, no such extremal graph is a complete join of three graphs of positive order. This gives a negative answer to a natural existence-only question motivated by the Simonovits Product Conjecture, in which one asks only for one product extremizer at each sufficiently large order.

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Chuandong Xu. 2026-08-16. A finite forbidden family with superlinear surplus and no three-factor product extremizers. https://arxiv.org/abs/2608.15777

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