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arXiv · 2606.20494

Strong non-principality of positive codegree Tur\'an density

Levente Bodnár·Jun Gao·Oleg Pikhurko·Mingyuan Rong·Shumin Sun
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Abstract

The \emph{minimum positive codegree} $\delta^+_{k-1}(G)$ of a $k$-graph $G$ is the minimum, over all $(k-1)$-sets that lie in at least one edge, of the number of edges containing that set. The \emph{positive codegree Tur\'an density} of a $k$-graph family $\mathcal{F}$ is the asymptotically maximum value of $\delta^+_{k-1}(G)/n$ over all $\mathcal{F}$-free $k$-graphs $G$ with $n\to\infty$ vertices. In this note, we establish a strong version of non-principality with respect to this density by proving that for every $k\ge3$ there exist two $k$-graphs $F_1$ and $F_2$ such that $$ 0<\gamma^+(F_1, F_2) < \min\{\gamma^+(F_1), \gamma^+(F_2)\}. $$

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Levente Bodnár, Jun Gao, Oleg Pikhurko, Mingyuan Rong, Shumin Sun. 2026-06-18. Strong non-principality of positive codegree Tur\'an density. https://arxiv.org/abs/2606.20494

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