Erd?s-Ko-Rado for three sets

Fix integers k?3 and n?3k/2. Let F be a family of k-sets of an n-element set so that whenever A,B,C∈F satisfy |A∪B∪C|?2k, we have A∩B∩C≠∅. We prove that with equality only when ?_F∈FF≠∅. This settles a conjecture of Frankl and Füredi [2], who proved the result for n?k²+3k.     

The maximum size of hypergraphs without generalized 4-cycles

Let fr(n) be the maximum number of edges in an r-uniform hypergraph on n vertices that does not contain four distinct edges A, B, C, D with A∪B=C∪D and A∩B=C∩D=∅. This problem was stated by Erd?s [P. Erd?s, Problems and results in combinatorial analysis, Congr. Numer. 19 (1977) 3-12]. It can be viewed as a generalization of the Turán problem for the 4-cycle to hypergraphs.Let . Füredi [Z. Füredi, Hypergraphs in which all disjoint pairs have distinct unions, Combinatorica 4 (1984) 161-168] observed that ?r?1 and conjectured that this is equality for every r?3. The best known upper bound ?r?3 was proved by Mubayi and Verstraëte [D. Mubayi, J. Verstraëte, A hypergraph extension of the bipartite Turán problem, J. Combin. Theory Ser. A 106 (2004) 237-253]. Here we improve this bound. Namely, we show that for every r?3, and ?₃?13/9. In particular, it follows that ?r→1 as r→∞.     

A new short proof of a theorem of Ahlswede and Khachatrian

Ahlswede and Khachatrian [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] proved the following theorem, which answered a question of Frankl and Füredi [P. Frankl, Z. Füredi, Nontrivial intersecting families, J. Combin. Theory Ser. A 41 (1986) 150-153]. Let 2?t+1?k?2t+1 and n?(t+1)(k−t+1). Suppose that F is a family of k-subsets of an n-set, every two of which have at least t common elements. If |_?F∈FF|<t, then , and this is best possible. We give a new, short proof of this result. The proof in [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] requires the entire machinery of the proof of the complete intersection theorem, while our proof uses only ordinary compression and an earlier result of Wilson [R.M. Wilson, The exact bound in the Erd?s-Ko-Rado theorem, Combinatorica 4 (1984) 247-257].     

A multiply intersecting Erd?s-Ko-Rado theorem — The principal case

Let m(n,k,r,t) be the maximum size of satisfying |F₁∩?∩F_r|≥t for all F₁,…,F_r∈F. We prove that for every p∈(0,1) there is some r₀ such that, for all r>r₀ and all t with 1≤t≤⌊(p^1−r−p)/(1−p)⌋−r, there exists n₀ so that if n>n₀ and p=k/n, then . The upper bound for t is tight for fixed p and r.     

Non-uniform Turán-type problems

Given positive integers n,k,t, with 2?k?n, and t<2^k, let m(n,k,t) be the minimum size of a family F of (nonempty distinct) subsets of [n] such that every k-subset of [n] contains at least t members of F, and every (k-1)-subset of [n] contains at most t-1 members of F. For fixed k and t, we determine the order of magnitude of m(n,k,t). We also consider related Turán numbers T_?r(n,k,t) and T_r(n,k,t), where T_?r(n,k,t) (T_r(n,k,t)) denotes the minimum size of a family such that every k-subset of [n] contains at least t members of F. We prove that T_?r(n,k,t)=(1+o(1))T_r(n,k,t) for fixed r,k,t with and n→∞.     

Small maximally disjoint union-free families

A family F of k-subsets of an n-set X is disjoint union-free (DUF) if all disjoint pairs of elements of F have distinct unions; that is, if for every A,B,C,D∈F, A∩B=C∩D=∅ and A∪B=C∪D implies {A,B}={C,D}. DUF families of maximum size have been studied by Erdös and Füredi. Let F be DUF with the property that F∪{E} is not DUF for any k-subset E of X not already in F. Then F is maximally DUF. We introduce the problem of finding the minimum size of maximally DUF families and provide bounds on this quantity for k=3.     

On the spectral radius of trees with fixed diameter

Let T_{(n, d)} be the set of trees on n vertices with diameter d. In this paper, the first spectral radii of trees in the set T_{(n, d)} (3 ? d ? n − 4) are characterized.     

Note on the 3-graph counting lemma

Szemerédi's regularity lemma proved to be a powerful tool in extremal graph theory. Many of its applications are based on the so-called counting lemma: if G is a k-partite graph with k-partition V₁∪?∪V_k, |V₁|=?=|V_k|=n, where all induced bipartite graphs G[V_i,V_j] are (d,ε)-regular, then the number of k-cliques K_k in G is . Frankl and Rödl extended Szemerédi's regularity lemma to 3-graphs and Nagle and Rödl established an accompanying 3-graph counting lemma analogous to the graph counting lemma above. In this paper, we provide a new proof of the 3-graph counting lemma.     

Heteroclinic solutions for a class of the second order Hamiltonian systems

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system , where q∈Rn and V∈C¹(R×Rn,R), V?0. We will assume that V and a certain subset M⊂Rn satisfy the following conditions. M is a set of isolated points and #M?2. For every sufficiently small ε>0 there exists δ>0 such that for all (t,z)∈R×Rn, if d(z,M)?ε then −V(t,z)?δ. The integrals , z∈M, are equi-bounded and −V(t,z)→∞, as |t|→∞, uniformly on compact subsets of Rn?M. Our result states that each point in M is joined to another point in M by a solution of our system.     

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that, for every locally compact abelian group G, the following statements are equivalent:

(i)

G contains no sequence such that {0}∪{±x_n∣n∈N} is infinite and quasi-convex in G, and x_n?0;

(ii)

one of the subgroups {g∈G∣2g=0} or {g∈G∣3g=0} is open in G;

(iii)

G contains an open compact subgroup of the form or for some cardinal κ.

     

EKR type inequalities for 4-wise intersecting families

Let 1?t?7 be an integer and let F be a k-uniform hypergraph on n vertices. Suppose that |A∩B∩C∩D|?t holds for all A,B,C,D∈F. Then we have if holds for some ε>0 and all n>n₀(ε). We apply this result to get EKR type inequalities for “intersecting and union families” and “intersecting Sperner families.”     

Structural properties of Riordan matrices and extending the matrices

We consider an infinite lower triangular matrix L=[?_n,k]_n,k∈N0 and a sequence Ω=(ω_n)_n∈N0 called the (a,b)-sequence such that every element ?_n+1,k+1 except lying in column 0 can be expressed as

     

A short proof of a cross-intersection theorem of Hilton

Families A₁,…,A_k of sets are said to be cross-intersecting if for any A_i∈A_i and A_j∈A_j, i≠j. A nice result of Hilton that generalises the Erd?s-Ko-Rado (EKR) Theorem says that if r≤n/2 and A₁,…,A_k are cross-intersecting sub-families of , then

     

Counting substructures I: Color critical graphs

Let F be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of F in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits (1968) [8], who proved that there is one copy of F, and of Rademacher, Erd?s (1962) [1] and [2] and Lovász and Simonovits (1983) [4], who proved similar counting results when F is a complete graph.One of the simplest cases of our theorem is the following new result. There is an absolute positive constant c such that if n is sufficiently large and 1?q<cn, then every n vertex graph with ⌊n²/4⌋+q edges contains at least

     

Stability of spacelike hypersurfaces in foliated spacetimes

Given a generalized Robertson-Walker spacetime whose warping function verifies a certain convexity condition, we classify strongly stable spacelike hypersurfaces with constant mean curvature. More precisely, we will show that given a closed, strongly stable spacelike hypersurface of with constant mean curvature H, if the warping function ? satisfying ?^″?max{H?^′,0} along M, then Mn is either maximal or a spacelike slice Mt₀={t₀}×F, for some t₀∈I.     

Universal adjacency matrices with two eigenvalues

Consider a graph Γ on n vertices with adjacency matrix A and degree sequence (d₁,…,d_n). A universal adjacency matrix of Γ is any matrix in Span {A,D,I,J} with a nonzero coefficient for A, where and I and J are the n×n identity and all-ones matrix, respectively. Thus a universal adjacency matrix is a common generalization of the adjacency, the Laplacian, the signless Laplacian and the Seidel matrix. We investigate graphs for which some universal adjacency matrix has just two eigenvalues. The regular ones are strongly regular, complete or empty, but several other interesting classes occur.