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)edges.This paper states spectral versions of the Erd6s-_Ko--Rado theorem:le...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)edges.This paper states spectral versions of the Erd6s-_Ko--Rado theorem:let G be an intersecting k-uniform hypergraph on n vertices with n≥2k.Then,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≤α<1,and Q^(*)(G)is the incidence Q-tensor,respectively.Furthermore,when n>2k,the extremal hypergraphs which attain the sharp upper bounds are characterized.The proof mainly relies on the Perron-Frobenius theorem for nonnegative tensor and the property of the maximizing connected hypergraphs.展开更多
A hypergraph H is an(n,m)-hypergraph if it contains n vertices and m hyperedges,where n≥1 and m≥0 are two integers.Let k be a positive integer and let L be a set of nonnegative integers.A hyper graph H is k-uniform ...A hypergraph H is an(n,m)-hypergraph if it contains n vertices and m hyperedges,where n≥1 and m≥0 are two integers.Let 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 L.In 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 L.We will provide some tight upper and lower bounds on k in terms of n,m and L.展开更多
A restricted signed r-set is a pair （A, f）, where A lohtain in [n] = {1, 2,…, n} is an r-set and f is a map from A to [n] with f（i） ≠ i for all i ∈ A. For two restricted signed sets （A, f） and （B, g）, we d...A restricted signed r-set is a pair （A, f）, where A lohtain in [n] = {1, 2,…, n} is an r-set and f is a map from A to [n] with f（i） ≠ i for all i ∈ A. For two restricted signed sets （A, f） and （B, g）, we define an order as （A, f） ≤ （B, g） if A C B and g｜A ： f A family .A of restricted signed sets on [n] is an intersecting antiehain if for any （A, f）, （B, g） ∈ A, they are incomparable and there exists x ∈ A ∩ B such that f（x） = g（x）. In this paper, we first give a LYM-type inequality for any intersecting antichain A of restricted signed sets, from which we then obtain ｜A｜≤ （r-1^n-1）（n-1）^r-1 if A. consists of restricted signed r-sets on [n]. Unless r = n = 3, equality holds if and only if A consists of all restricted signed r-sets （A, f） such that x0∈ A and f（x0） =ε0 for some fixed x0 ∈ [n], ε0 ∈ [n] ／ {x0}.展开更多
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, w...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.展开更多
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...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.展开更多
基金the National Natural Science Foundation of China(Nos.11971311,11531001)the Montenegrin-Chinese Science and Technology Cooperation Project(No.3-12).
文摘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)edges.This paper states spectral versions of the Erd6s-_Ko--Rado theorem:let G be an intersecting k-uniform hypergraph on n vertices with n≥2k.Then,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≤α<1,and Q^(*)(G)is the incidence Q-tensor,respectively.Furthermore,when n>2k,the extremal hypergraphs which attain the sharp upper bounds are characterized.The proof mainly relies on the Perron-Frobenius theorem for nonnegative tensor and the property of the maximizing connected hypergraphs.
基金Supported by National Natural Science Foundation of China(Grant Nos.12242111,12131013,12171393,12071370,71973103,U1803263,11601430)Natural Science Foundation of Shaanxi Province(Grant Nos.2021JM-040,2020JQ-099)+2 种基金Shaanxi Fundamental Science Research Project for Mathematics and Physics(Grant No.22JSZ009)Guangdong Basic and Applied Basic Research Foundation(Grant Nos.2023A1515030208,2022A1515010899)the Fundamental Research Funds for the Central Universities。
文摘A hypergraph H is an(n,m)-hypergraph if it contains n vertices and m hyperedges,where n≥1 and m≥0 are two integers.Let 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 L.In 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 L.We will provide some tight upper and lower bounds on k in terms of n,m and L.
基金Supported by the doctoral Foundation of Yanshan University(No.B314)
文摘A restricted signed r-set is a pair （A, f）, where A lohtain in [n] = {1, 2,…, n} is an r-set and f is a map from A to [n] with f（i） ≠ i for all i ∈ A. For two restricted signed sets （A, f） and （B, g）, we define an order as （A, f） ≤ （B, g） if A C B and g｜A ： f A family .A of restricted signed sets on [n] is an intersecting antiehain if for any （A, f）, （B, g） ∈ A, they are incomparable and there exists x ∈ A ∩ B such that f（x） = g（x）. In this paper, we first give a LYM-type inequality for any intersecting antichain A of restricted signed sets, from which we then obtain ｜A｜≤ （r-1^n-1）（n-1）^r-1 if A. consists of restricted signed r-sets on [n]. Unless r = n = 3, equality holds if and only if A consists of all restricted signed r-sets （A, f） such that x0∈ A and f（x0） =ε0 for some fixed x0 ∈ [n], ε0 ∈ [n] ／ {x0}.
文摘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.
基金supported by FWO-Vlaanderen(Research Foundation-Flanders)
文摘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.
