Shadows and intersections in vector spaces

We prove a vector space analog of a version of the Kruskal-Katona theorem due to Lovász. We apply this result to extend Frankl's theorem on r-wise intersecting families to vector spaces. In particular, we obtain a short new proof of the Erd?s-Ko-Rado theorem for vector spaces.     

The Erd?s-Ko-Rado properties of set systems defined by double partitions

Let F be a family of subsets of a finite set V. The star ofFatv∈V is the sub-family {A∈F:v∈A}. We denote the sub-family {A∈F:|A|=r} by F^(r).A double partitionP of a finite set V is a partition of V into large sets that are in turn partitioned into small sets. Given such a partition, the family F(P)induced byP is the family of subsets of V whose intersection with each large set is either contained in just one small set or empty.Our main result is that, if one of the large sets is trivially partitioned (that is, into just one small set) and 2r is not greater than the least cardinality of any maximal set of F(P), then no intersecting sub-family of F(P)^(r) is larger than the largest star of F(P)^(r). We also characterise the cases when every extremal intersecting sub-family of F(P)^(r) is a star of F(P)^(r).     

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.”     

Partitioning Boolean lattices into antichains

Let f(n) be the smallest integer t such that a poset obtained from a Boolean lattice with n atoms by deleting both the largest and the smallest elements can be partitioned into t antichains of the same size except for possibly one antichain of a smaller size. In this paper, it is shown that f(n)b n²/log n. This is an improvement of the best previously known upper bound for f(n).     

The Erdős-Ko-Rado theorem for finite affine spaces

In [W.N. Hsieh, Intersection theorems for finite vector spaces, Discrete Math. 12 (1975) 1–16], Hsieh obtained the Erd?s-Ko-Rado theorem for finite vector spaces. This paper generalizes Hsieh’s result and obtains the Erd?s-Ko-Rado theorem for finite affine spaces.     

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 .     

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.     

An Erd?s-Ko-Rado theorem for partial permutations

Let [n] denote the set of positive integers {1,2,…,n}. An r-partial permutation of [n] is a pair (A,f) where A⊆[n], |A|=r and f:A→[n] is an injective map. A set A of r-partial permutations is intersecting if for any (A,f), (B,g)∈A, there exists x∈A∩B such that f(x)=g(x). We prove that for any intersecting family A of r-partial permutations, we have .It seems rather hard to characterize the case of equality. For 8?r?n-3, we show that equality holds if and only if there exist x₀ and ε₀ such that A consists of all (A,f) for which x₀∈A and f(x₀)=ε₀.     

Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs

Brouwer, Godsil, Koolen and Martin [Width and dual width of subsets in polynomial association schemes, J. Combin. Theory Ser. A 102 (2003) 255-271] introduced the width w and the dual width w^* of a subset in a distance-regular graph and in a cometric association scheme, respectively, and then derived lower bounds on these new parameters. For instance, subsets with the property w+w^*=d in a cometric distance-regular graph with diameter d attain these bounds. In this paper, we classify subsets with this property in Grassmann graphs, bilinear forms graphs and dual polar graphs. We use this information to establish the Erd?s-Ko-Rado theorem in full generality for the first two families of graphs.     

Erdős–Ko–Rado and Hilton–Milner theorems for two-forms

The celebrated Erd?s–Ko–Rado theorem about the maximal size of an intersecting family of r-element subsets of $\{1,\dots ,n\}$ was extended to the setting of exterior algebra in [5, Theorem 2.3] and in [6, Theorem 1.4]. However, the equality case has not been settled yet. In this short note, we show that the extension of the Erd?s–Ko–Rado theorem and the characterization of the equality case therein, as well as those of the Hilton–Milner theorem to the setting of exterior algebra in the simplest non-trivial case of two-forms follow from a folklore puzzle about possible arrangements of an intersecting family of lines.     

Extremal t-intersecting families for direct products

In this paper, by shifting technique we study t-intersecting families for direct products where the ground set is divided into several parts. Assuming the size of each part is sufficiently large, we determine all extremal t-intersecting families for direct products. We also prove that every largest t-intersecting subfamily of a more general family introduced by Katona is trivial under certain conditions.     

Erdos-Ko-Rado Theorems of Labeled Sets

For k = (k1, ··· , kn) ∈ Nn, 1 ≤ k1 ≤···≤ kn, let Lkr be the family of labeled r-sets on k given by Lkr := {{(a1, la1), ··· , (ar, lar)} : {a1, ··· , ar} ■[n],lai ∈ [kai],i = 1, ··· , r}. A family A of labeled r-sets is intersecting if any two sets in A intersect. In this paper we give the sizes and structures of intersecting families of labeled r-sets.     

Erd?s-Ko-Rado theorem for irreducible imprimitive reflection groups

Let Ω be a finite set, and let G be a permutation group on Ω. A subset H of G is called intersecting if for any σ, π ∈ H, they agree on at least one point. We show that a maximal intersecting subset of an irreducible imprimitive reflection group G(m, p, n) is a coset of the stabilizer of a point in {1, …, n} provided n is sufficiently large.     

The Erd?s-Ko-Rado properties of various graphs containing singletons

Let G=(V,E) be a graph. For r≥1, let be the family of independent vertex r-sets of G. For v∈V(G), let denote the star. G is said to be r-EKR if there exists v∈V(G) such that for any non-star family A of pair-wise intersecting sets in . If the inequality is strict, then G is strictlyr-EKR.Let Γ be the family of graphs that are disjoint unions of complete graphs, paths, cycles, including at least one singleton. Holroyd, Spencer and Talbot proved that, if G∈Γ and 2r is no larger than the number of connected components of G, then G is r-EKR. However, Holroyd and Talbot conjectured that, if G is any graph and 2r is no larger than μ(G), the size of a smallest maximal independent vertex set of G, then G is r-EKR, and strictly so if 2r<μ(G). We show that in fact, if G∈Γ and 2r is no larger than the independence number of G, then G is r-EKR; we do this by proving the result for all graphs that are in a suitable larger set Γ^′?Γ. We also confirm the conjecture for graphs in an even larger set Γ^″?Γ^′.     

Theorems of Erd?s-Ko-Rado type in polar spaces

We consider Erd?s-Ko-Rado sets of generators in classical finite polar spaces. These are sets of generators that all intersect non-trivially. We characterize the Erd?s-Ko-Rado sets of generators of maximum size in all polar spaces, except for H(4n+1,q²) with n?2.