Bounding the Independence Number in Some $(n,k,\ell ,\lambda )$-Hypergraphs

Abstract

Given positive integers k, $\ell$ and $\lambda$ such that $2\leqslant \ell \leqslant k-1$, an $(n,k,\ell ,\lambda )$-hypergraph $\mathcal {H}$ is a k-uniform hypergraph on the vertex set [n] in which every subset of size $\ell$ is contained in at most $\lambda$ edges. The independence number $\alpha (\mathcal {H})$ of $\mathcal {H}$ is the maximum size of a subset of vertices which contains no edges of $\mathcal {H}$. Let $f(n,k,\ell ,\lambda )=\min \{\alpha (\mathcal {H})\mid \mathcal {H}\ \mathrm{{is\ an\ }}(n,k,\ell ,\lambda )$-hypergraph$\}$. In this paper we show that for any given positive integers $k\geqslant 5$, $\frac{2k+4}{5}<\ell \leqslant k-2$ and $\lambda \leqslant \left\lfloor \dfrac{n^{\frac{5\ell -2k-4}{3k-9}}}{\omega (n)}\right\rfloor$,

$$\begin{aligned} f(n,k,\ell ,\lambda )\geqslant C_{k,\ell }\left( \frac{n}{\lambda }\log \frac{n}{\lambda }\right) ^{\frac{1}{\ell }}, \end{aligned}$$

where $\omega (n)\rightarrow \infty$ arbitrarily slowly as $n\rightarrow \infty$ and $C_{k,\ell }$ is a constant depending only on k and $\ell$. In particular, $C_{k,\ell }\sim \frac{\ell -1}{e}$ as $k\rightarrow \infty$. It generalizes the results from the case $\ell =k-1$ or $\lambda =1$. An upper bound on $f(n,k,\ell ,\lambda )$ is also obtained for $k\geqslant 3$, $2\leqslant \ell \leqslant k-1$ and $\log n\ll \lambda \ll n$, by considering a random k-uniform hypergraph $\mathcal {H}_k(n,p)$.