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arXiv · 2608.19923

On the Generalized Rational Exponents Conjecture

Jianfeng Hou·Caihong Yang
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Abstract

For fixed graphs $H$ and $F$, let $\ex(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. In this note, we prove the generalized rational exponents conjecture, posed by Gerbner and Palmer, showing that for every rational number $\alpha\ge1$, there exist fixed graphs $H_\alpha$ and $F_\alpha$ such that \[ \ex(n,H_\alpha,F_\alpha)=\Theta(n^\alpha). \] Furthermore, the counting graph $H_\alpha$ can always be chosen connected with diameter at most $3$. Our argument hinges on a localization--compression--shift framework, which transforms the Bukh--Conlon finite family construction for edges into a generalized Tur\'an problem setting with a single forbidden graph.

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Jianfeng Hou, Caihong Yang. 2026-08-20. On the Generalized Rational Exponents Conjecture. https://arxiv.org/abs/2608.19923

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