Turán numbers of $4$-uniform tight even cycles minus one edge

Abstract

For every integer $k \ge 1$ and sufficiently large $n$, we show that the extremal construction for the Turán number of the $4$-uniform tight cycle of length $4k+2$ minus one edge is a complete odd-bipartite $4$-graph. In particular, since $C_{6}^{4-}$ contains the $4$-uniform expanded triangle as a subgraph, our result extends that of Frankl and Keevash--Sudakov on the Turán density and the Turán number of the $4$-uniform expanded triangle. We also show that the Turán density of $C_{4k+2}^{4}$ is $1/2$ for all integers $k \ge 2$, and establish the corresponding stability result. This strengthens the result of Sankar on the Turán density of $C_{4k+2}^{4}$ which holds only for all sufficiently large $k$.