Hypergraphs with no tight cycles

Abstract

We show that every r r -uniform hypergraph on n n vertices which does not contain a tight cycle has at most O ( n r − 1 ( log ⁡ n ) 5 ) O(n^{r-1} (\log n)^5) edges. This is an improvement on the previously best-known bound, of n r − 1 e O ( log ⁡ n ) n^{r-1} e^{O(\sqrt {\log n})} , due to Sudakov and Tomon, and our proof builds on their work. A recent construction of B. Janzer implies that our bound is tight up to an O ( ( log ⁡ n ) 4 log ⁡ log ⁡ n ) O((\log n)^4 \log \log n) factor.