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.”     

Setwise intersecting families of permutations

A family of permutations A⊂Sn is said to be t-set-intersecting if for any two permutations σ,π∈A, there exists a t-set x whose image is the same under both permutations, i.e. σ(x)=π(x). We prove that if n is sufficiently large depending on t, the maximum-sized t-set-intersecting families of permutations in Sn are cosets of stabilizers of t-sets. The t=2 case of this was conjectured by János Körner. It can be seen as a variant of the Deza-Frankl conjecture, proved in Ellis, Friedgut and Pilpel (2011) [3]. Our proof uses similar techniques to those of Ellis, Friedgut and Pilpel (2011) [3], namely, eigenvalue methods, together with the representation theory of the symmetric group, but the combinatorial part of the proof is harder.     

Isols and maximal intersecting classes

In transfinite arithmetic 2ⁿ is defined as the cardinality of the family of all subsets of some set v with cardinality n. However, in the arithmetic of recursive equivalence types (RETs) 2^N is defined as the RET of the family of all finite subsets of some set v of nonnegative integers with RET N. Suppose v is a nonempty set. S is a class over v, if S consists of finite subsets of v and has v as its union. Such a class is an intersecting class (IC) over v, if every two members of S have a nonempty intersection. An IC over v is called a maximal IC (MIC), if it is not properly included in any IC over v. It is known and readily proved that every MIC over a finite set v of cardinality n ≥ 1 has cardinality 2^n-1. In order to generalize this result we introduce the notion of an ω-MIC over v. This is an effective analogue ot the notion of an MIC over v such that a class over a finite set v is an ω-MIC iff it is an MIC. We then prove that every ω-MIC over an isolated set v of RET N ≥ 1 has RET 2^N-1. This is a generalization, for while there only are χ₀ finite sets, there are ? isolated sets, where c denotes the cardinality of the continuum, namely all the finite sets and the c immune sets. MSC: 03D50.     

Listing minimal edge-covers of intersecting families with applications to connectivity problems

Let G=(V,E) be a directed/undirected graph, let s,t∈V, and let F be an intersecting family on V (that is, X∩Y,X∪Y∈F for any intersecting X,Y∈F) so that s∈X and t∉X for every X∈F. An edge set I⊆E is an edge-cover of F if for every X∈F there is an edge in I from X to V−X. We show that minimal edge-covers of F can be listed with polynomial delay, provided that, for any I⊆E the minimal member of the residual family F_I of the sets in F not covered by I can be computed in polynomial time. As an application, we show that minimal undirected Steiner networks, and minimal k-connected and k-outconnected spanning subgraphs of a given directed/undirected graph, can be listed in incremental polynomial time.     

Families intersecting on an interval

We shall be interested in the following Erd?s-Ko-Rado-type question. Fix some set B⊂[n]={1,2,…,n}. How large a subfamily A of the power set P[n] can we find such that the intersection of any two sets in A contains a cyclic translate (modulo n) of B? Chung, Graham, Frankl and Shearer have proved that, in the case where B=[t] is a block of length t, we can do no better than taking A to consist of all supersets of B. We give an alternative proof of this result, which is in a certain sense more ‘direct’.     

Minimum size transversals in uniform hypergraphs

A set of vertices in a hypergraph which meets all the edges is called a transversal. The transversal number τ(H)$\tau \left(H\right)$ of a hypergraph H$H$ is the minimum cardinality of a transversal in H$H$. A classical greedy algorithm for constructing a transversal of small size selects in each step a vertex which has the largest degree in the hypergraph formed by the edges not met yet. The analysis of this algorithm (by Chvátal and McDiarmid (1992)  [3]) gave some upper bounds for τ(H)$\tau \left(H\right)$ in a uniform hypergraph H$H$ with a given number of vertices and edges. We discuss a variation of this greedy algorithm. Analyzing this new algorithm, we obtain upper bounds for τ(H)$\tau \left(H\right)$ which improve the bounds by Chvátal and McDiarmid.     

On a question of Sós about 3‐uniform friendship hypergraphs

The well‐known Friendship Theorem states that if G is a graph in which every pair of vertices has exactly one common neighbor, then G has a single vertex joined to all others (a “universal friend”). V. Sós defined an analogous friendship property for 3‐uniform hypergraphs, and gave a construction satisfying the friendship property that has a universal friend. We present new 3‐uniform hypergraphs on 8, 16, and 32 vertices that satisfy the friendship property without containing a universal friend. We also prove that if n ≤ 10 and n ≠ 8, then there are no friendship hypergraphs on n vertices without a universal friend. These results were obtained by computer search using integer programming. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 253–261, 2008     

Uniform stability of resolvent families

In this article we study uniform stability of resolvent families associated to an integral equation of convolution type. We give sufficient conditions for the uniform stability of the resolvent family in Hilbert and Banach spaces. Our main result can be viewed as a substantial generalization of the Gearhart-Greiner-Prüss characterization of exponential stability for strongly continuous semigroups.

     

On cross t-intersecting families of sets

For all p, t with 0<p<0.11 and 1?t?1/(2p), there exists n₀ such that for all n, k with n>n₀ and k/n=p the following holds: if A and B are k-uniform families on n vertices, and |A∩B|?t holds for all A∈A and B∈B, then .     

Exponential trichotomy and p-admissibility for evolution families on the real line

The aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of evolution families on the real line. We prove that if p ∈ (1,∞) and the pair (C_b(R,X),C_c(R,X)) is uniformly p-admissible for an evolution family ={U(t,s)}_t≥_s then is uniformly exponentially trichotomic. After that we analyze when the uniform p-admissibility of the pair (C_b(R, X), C_c(R, X)) becomes a necessary condition for uniform exponential trichotomy. As applications of these results we study the uniform exponential dichotomy of evolution families. We obtain that in certain conditions, the admissibility of the pair (C_b(R,X),L^p(R,X)) for an evolution family ={U(t,s)}_t≥_s is equivalent with its uniform exponential dichotomy.     

On the number of minimal transversals in 3-uniform hypergraphs

We prove that the number of minimal transversals (and also the number of maximal independent sets) in a 3-uniform hypergraph with n vertices is at most cⁿ, where c≈1.6702. The best known lower bound for this number, due to Tomescu, is adⁿ, where d=10^1/5≈1.5849 and a is a constant.     

Disjoint perfect matchings in 3‐uniform hypergraphs

For a hypergraph H, let denote the minimum vertex degree in H. Kühn, Osthus, and Treglown proved that, for any sufficiently large integer n with , if H is a 3‐uniform hypergraph with order n and then H has a perfect matching, and this bound on is best possible. In this article, we show that under the same conditions, H contains at least pairwise disjoint perfect matchings, and this bound is sharp.     

Judiciously 3‐partitioning 3‐uniform hypergraphs

Bollobás, Reed, and Thomason proved every 3‐uniform hypergraph ? with m edges has a vertex‐partition V()=V₁?V₂?V₃ such that each part meets at least edges, later improved to 0.6m by Halsegrave and improved asymptotically to 0.65m+o(m) by Ma and Yu. We improve this asymptotic bound to , which is best possible up to the error term, resolving a special case of a conjecture of Bollobás and Scott.