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

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.

Disclosure

“Statement of AI use. The proof strategy and initial manuscript draft were generated by GPT-5.6 Sol in response to a problem formulated by the author. The author verified and revised the manuscript and assumes responsibility for all content. 2 Auxiliary results 2.1 Classical extremal inputs We use three classical extremal i”

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Classification

Substantial proof generation

Multiplier

10

Verified

2026-08-23T05:17:53.690079Z

Structural counts