Strong non-principality of positive codegree Turán density

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 Turán 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 $k\ge 3$ there exist two $k$-graphs $F_1$ and $F_2$ such that $$0<{\gamma }^{+}(F_1,F_2)<\min \{{\gamma }^{+}(F_1),{\gamma }^{+}(F_2)\}.$$

7 pages