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Uniform Hypergraphs under Certain Intersection Constraints between Hyperedges

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Acta Mathematica Sinica,English Series 2023年第6期39卷 1153-1170页

作者： Yan Dong BAI Bin Long LI Jiu Qiang LIU Sheng Gui ZHANG School of Mathematics and Statistics Northwestern Polytechnical UniversityXi'an 710072P.R.China School of Mathematical Sciences Xi'an University of Finance and EconomicsXi'an 710100P.R.China Department of Mathematics Eastern Michigan UniversityYpsilantiMI 48197USA

A hypergraph H is an(n,m)-hypergraph if it contains n vertices and m hyperedges,where n≥1 and m≥0 are two *** k be a positive integer and let L be a set of nonnegative integers.A hyper graph H is k-uniform if all its hyperedges have the same size k,and H is L-intersecting if the number of common vertices of every two hyperedges belongs to *** this paper,we propose and investigate the problem of estimating the maximum k among all k-uniform L-intersecting(n,m)-hypergraphs for fixed n,m and *** will provide some tight upper and lower bounds on k in terms of n,m and L.

关键词： Uniform hypergraph erdos-ko-rado theorem extremal set theory

A Common Generalization to theorems on Set Systems with L-intersections

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Acta Mathematica Sinica,English Series 2018年第7期34卷 1087-1100页

作者： Jiu Qiang LIU Sheng Gui ZHANG Ji Meng XIAO School of Management Engineering Xi'an University of Finance and Economics Xi'an 710100 P. R. China Department of Mathematics Eastern Michigan University Ypsilanti MI 48197 USA Department of Mathematics Northwestern Polytechnical University Xi'an 710072 P. R. China

In this paper, we provide a common generalization to the well-known erdos-ko-rado theorem, Frankl-Wilson theorem, Alon-Babai-Suzuki theorem, and Snevily theorem on set systems with L-intersections. As a consequence, we derive a result which strengthens substantially the well- known theorem on set systems with k-wise E-intersections by Furedi and Sudakov [J. Combin. Theory, Set. A, 105, 143-159 （2004）]. We will also derive similar results on E-intersecting families of subspaces of an n-dimensional vector space over a finite field Fq, where q is a prime power.

关键词： Alon-Babai-Suzuki theorem erdos-ko-rado theorem Frankl-Wilson theorem Snevily theorem multilinear polynomials

The Spectral Radii of Intersecting Uniform Hypergraphs

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Communications on Applied Mathematics and Computation 2021年第2期3卷 243-256页

作者： Peng-Li Zhang Xiao-Dong Zhang School of Mathematical Sciences MOE-LSCSHL-MACShanghai Jiao Tong UniversityShanghai 200240China

The celebrated erdos-ko-rado theorem states that given n≥2k,every intersecting k-uni-n-1 form hypergraph G on n vertices has at most(n-1 k-1)*** paper states spectral versions of the Erd6s-_ko--rado theorem:let G be an intersecting k-uniform hypergraph on n vertices with n≥***,the sharp upper bounds for the spectral radius of Aα(G)and 2*(G)are presented,where Aα(G)=αD(G)+(1-α).A(G)is a convex linear combination of the degree diagonal tensor D(G)and the adjacency tensor A(G)for 0≤α2k,the extremal hypergraphs which attain the sharp upper bounds are *** proof mainly relies on the Perron-Frobenius theorem for nonnegative tensor and the property of the maximizing connected hypergraphs.

关键词： erdos-ko-rado theorem Intersecting hypergraph Tensor Spectral radius

theorems of Erds-ko-rado type in geometrical settings

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Science China Mathematics 2013年第7期56卷 1333-1348页

作者： DE BOECK Maarten STORME Leo Department of Mathematics Krijgslaan 281-Building S22 9000 GentFlanders Belgium

The original erdos-ko-rado problem has inspired much research. It started as a study on sets of pairwise intersecting k-subsets in an n-set, then it gave rise to research on sets of pairwise non-trivially intersecting k-dimensional vector spaces in the vector space V（n, q） of dimension n over the finite field of order q, and then research on sets of pairwise non-trivially intersecting generators and planes in finite classical polar spaces. We summarize the main results on the erdos-ko-rado problem in these three settings, mention the ErdSs-ko-rado problem in other related settings, and mention open problems for future research.

关键词： erdos-ko-rado theorem finite sets finite vector spaces finite classical polar spaces