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

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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 \(\mathcal {H}_k(n,p)\).

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Acknowledgements

We are immensely grateful to the anonymous reviewer for his/her detailed and helpful comments on an earlier version of our manuscript. Without his/her help, it is impossible for us to write this manuscript in its current form. The first author Fang Tian was partially supported by the NFSC (No. 11101256) and CSC [2017]3192, who is now a visiting research fellow in Australian National University.

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Authors and Affiliations

  1. Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Fang Tian

  2. School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Zi-Long Liu

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  1. Fang Tian
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  2. Zi-Long Liu
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Correspondence to Fang Tian.

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The work was partially supported by NSFC (No. 11101256).

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Tian, F., Liu, ZL. Bounding the Independence Number in Some \((n,k,\ell ,\lambda )\)-Hypergraphs. Graphs and Combinatorics 34, 845–861 (2018). https://doi.org/10.1007/s00373-018-1911-y

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