A set system is called t-intersecting if every two members meet each other in at least t elements. Katona determined the minimum ratio of the shadow and the size of such families and showed that the Erdős–Ko–Rado theorem immediately follows from this result. The aim of this note is to reproduce the proof to obtain a slight improvement in the Kneser graph. We also give a brief overview of corres...

Let k and m be positive integers. A collection of k-multisets from {1, . . . ,m} is intersecting if every pair of multisets from the collection is intersecting. We prove that for m ≥ k +1, the size of the largest such collection is ( m+k−2 k−1 ) and that when m > k + 1, only a collection of all the k-multisets containing a fixed element will attain this bound. The size and structure of the larg...

Chih-Ko Yeh,1,2 Tazuko K. Hymer,1 April L. Sousa,1 Bin-Xian Zhang,3,4 Meyer D. Lifschitz,3,4 and Michael S. Katz1,4 1Geriatric Research, Education and Clinical Center, and 3Research Service, South Texas Veterans Health Care System, Audie L. Murphy Division, San Antonio 78229-4404; and Departments of 2Dental Diagnostic Science and 4Medicine, University of Texas Health Science Center at San Anton...

A family of sets is t-intersecting if any two sets from the family contain at least t common elements. Given a t-intersecting family of r-sets from an n-set, how many distinct sets of size k can occur as pairwise intersections of its members? We prove an asymptotic upper bound on this number that can always be achieved. This result can be seen as a generalization of the Erdős-Ko-Rado theorem.

In this paper we study a question related to the classical Erdős–Ko–Rado theorem, which states that any family of k-element subsets of the set [n] = {1, . . . , n} in which any two sets intersect, has cardinality at most (︀ n−1 k−1 )︀ . We say that two non-empty families are A,B ⊂ (︀[n] k )︀ are s-cross-intersecting, if for any A ∈ A, B ∈ B we have |A ∩ B| ≥ s. In this paper we determine the ma...

We show that Ossa’s theorem splitting ku∧BV for elementary abelian groups V follows from general facts about ku∧BZ/2 and Adams covers. For completeness, we also provide the analogous results for ko ∧BV .

A cross-intersecting Erdős-Ko-Rado set of generators of a finite classical polar space is a pair (Y,Z) of sets of generators such that all y ∈ Y and z ∈ Z intersect in at least a point. We provide upper bounds on |Y | · |Z| and classify the crossintersecting Erdős-Ko-Rado sets of maximum size with respect to |Y | · |Z| for all polar spaces except some Hermitian polar spaces.

Ahlswede and Khachatrian's diametric theorem is a weighted version of their complete intersection theorem, which is itself a well known extension of the t-intersecting Erd˝ os-Ko-Rado theorem. The complete intersection theorem says that the maximum size of a family of subsets of [n] = {1,. .. , n}, every pair of which intersects in at least t elements, is the size of certain trivially intersect...

Let S(n) be the symmetric group on n points. A subset S of S(n) is intersecting if for any pair of permutations π, σ in S there is a point i ∈ {1, . . . , n} such that π(i) = σ(i). Deza and Frankl [9] proved that if S ⊆ S(n) is intersecting then |S| ≤ (n− 1)!. Further, Cameron and Ku [4] show that the only sets that meet this bound are the cosets of a stabilizer of a point. In this paper we giv...

1. Two vector bundles E and F over the finite connected CW complex X are /-equivalent, if their sphere-bundles S(E) and S(F) are of the same fiber-homotopy type. If they become /-equivalent after a suitable number of trivial bundles is added to both of them they are stably /-equivalent. This note concerns itself with a new stable /-invariant 6(E), which was suggested by the recent work of Atiya...