The Erdös-Ko-Rado theorem for twisted Grassmann graphs

We present a “modern” approach to the Erdös-Ko-Rado theorem for Q-polynomial distance-regular graphs and apply it to the twisted Grassmann graphs discovered in 2005 by Van Dam and Koolen.     

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.     

The exact bound in the Erdös-Ko-Rado theorem

This paper contains a proof of the following result: ifn≧(t+1)(k?t?1), then any family ofk-subsets of ann-set with the property that any two of the subsets meet in at leastt points contains at most \(\left( {\begin{array}{*{20}c} {n - t} \\ {k - t} \\ \end{array} } \right)\) subsets. (By a theorem of P. Frankl, this was known whent≧15.) The bound (t+1)(k-t-1) represents the best possible strengthening of the original 1961 theorem of Erdös, Ko, and Rado which reaches the same conclusion under the hypothesisn≧t+(k?t) \(\left( {\begin{array}{*{20}c} k \\ t \\ \end{array} } \right)^3 \) . Our proof is linear algebraic in nature; it may be considered as an application of Delsarte’s linear programming bound, but somewhat lengthy calculations are required to reach the stated result. (A. Schrijver has previously noticed the relevance of these methods.) Our exposition is self-contained.     

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.     

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.     

An Erdös-Ko-Rado theorem for restricted signed sets

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}.     

An Erdős-Ko-Rado theorem for finite classical polar spaces

Consider a finite classical polar space of rank \(d\ge 2\) and an integer n with \(0<n<d\). In this paper, it is proved that the set consisting of all subspaces of rank n that contain a given point is a largest Erd?s-Ko-Rado set of subspaces of rank n of the polar space. We also show that there are no other Erd?s-Ko-Rado sets of subspaces of rank n of the same size.     

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^′.     

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.     

Erd?s-Ko-Rado theorems for simplicial complexes

A recent framework for generalizing the Erd?s-Ko-Rado theorem, due to Holroyd, Spencer, and Talbot, defines the Erd?s-Ko-Rado property for a graph in terms of the graph's independent sets. Since the family of all independent sets of a graph forms a simplicial complex, it is natural to further generalize the Erd?s-Ko-Rado property to an arbitrary simplicial complex. An advantage of working in simplicial complexes is the availability of algebraic shifting, a powerful shifting (compression) technique, which we use to verify a conjecture of Holroyd and Talbot in the case of sequentially Cohen-Macaulay near-cones.     

An Erdös-Ko-Rado theorem for the subcubes of a cube

LetP be that partially ordered set whose elements are vectors x=(x ₁, ...,x _n ) withx _i ε {0, ...,k} (i=1, ...,n) and in which the order is given byx≦y iffx _i =y _i orx _i =0 for alli. LetN _i (P)={x εP : |{j:x _j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsx ≮y, x ≯ y, and there exists az εN ₁(P) such thatz≦x andz≦y. Let ? be the set of all vectorsf=(f ₀, ...,f _n ) for which there is an Erdös-Ko-Rado familyF inP such that |N _i (P) ∩F|=f _i (i=0, ...,n) and let 〈?〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and \(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈?〉.     

The Erd?s-Ginzburg-Ziv theorem for finite solvable groups

Let G be a non-cyclic finite solvable group of order n, and let S=(g₁,…,g_k) be a sequence of k elements (repetition allowed) in G. In this paper we prove that if , then there exist some distinct indices i₁,i₂,…,i_n such that the product g_i1g_i2?g_in=1. This result substantially improves the Erd?s-Ginzburg-Ziv theorem and other existing results.     

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).     

The de Bruijn–Erd?s theorem for hypergraphs

Fix integers n ≥ r ≥ 2. A clique partition of ${{[n] \choose r}}$ is a collection of proper subsets ${A_1, A_2, \ldots, A_t \subset [n]}$ such that ${\bigcup_i{A_i \choose r}}$ is a partition of ${{[n]\choose r}}$ . Let cp(n, r) denote the minimum size of a clique partition of ${{[n] \choose r}}$ . A classical theorem of de Bruijn and Erd?s states that cp(n, 2) =?n. In this paper we study cp(n, r), and show in general that for each fixed r ≥ 3, $${\rm cp}(n, r) \geq (1 + o(1))n^{r/2} \quad \quad {\rm as} \, \, n \rightarrow \infty.$$ We conjecture cp(n, r) =?(1 +?o(1))n ^r/2. This conjecture has already been verified (in a very strong sense) for r = 3 by Hartman–Mullin–Stinson. We give further evidence of this conjecture by constructing, for each r ≥ 4, a family of (1?+?o(1))n ^r/2 subsets of [n] with the following property: no two r-sets of [n] are covered more than once and all but o(n ^r ) of the r-sets of [n] are covered. We also give an absolute lower bound ${{\rm cp}(n, r) \geq {n \choose r}/{q + r - 1 \choose r}}$ when n =?q ² + q +?r ? 1, and for each r characterize the finitely many configurations achieving equality with the lower bound. Finally we note the connection of cp(n, r) to extremal graph theory, and determine some new asymptotically sharp bounds for the Zarankiewicz problem.     

The Erdős-Szekeres theorem and congruences

The following problem of combinatorial geometry is considered. Given positive integers n and q, find or estimate a minimal number h for which any set of h points in general position in the plane contains n vertices of a convex polygon for which the number of interior points is divisible by q. For a wide range of parameters, the existing bound for h is dramatically improved.