On linear $k$-graphs with codegree Turán density arbitrarily close to zero

Abstract

Let $F$ be a $k$-uniform hypergraph, abbreviated as $k$-graph. The codegree Turán density $\ddot{I}{€}_{\mathrm{co}}(F)$ is the supremum over all ${\hat{I}}^3\in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least ${\hat{I}}^{3}n$ edges. In this paper, we prove that there is a linear $k$-graph $F$ with $0<\ddot{I}{€}_{co}(F)<\epsilon$ for any $\epsilon >0$. The special case $k=3$ solve a question proposed by Ding, Lamaison, Liu, Wang and Yang (JLMS, 2025). The main method combines an affine-plane-type incidence structure over a finite field and elementary number-theoretic arguments.