Open Access Publications from the University of California

## Published Web Location

https://doi.org/10.5070/C61055374
Abstract

An $$r$$-uniform tight cycle of length $$\ell>r$$ is a hypergraph with vertices $$v_1,\dots,v_\ell$$ and edges $$\{v_i,v_{i+1},\dots,v_{i+r-1}\}$$ (for all $$i$$), with the indices taken modulo $$\ell$$. It was shown by Sudakov and Tomon that for each fixed $$r\geq 3$$, an $$r$$-uniform hypergraph on $$n$$ vertices which does not contain a tight cycle of any length has at most $$n^{r-1+o(1)}$$ hyperedges, but the best known construction (with the largest number of edges) only gives $$\Omega(n^{r-1})$$ edges. In this note we prove that, for each fixed $$r\geq 3$$, there are $$r$$-uniform hypergraphs with $$\Omega(n^{r-1}\log n/\log\log n)$$ edges which contain no tight cycles, showing that the $$o(1)$$ term in the exponent of the upper bound is necessary.

Mathematics Subject Classifications: 05C65, 05C38