Let d and t be fixed positive integers, and let K t,...,t denote the complete d-partite hypergraph with t vertices in each of its parts, whose hyperedges are the d-tuples of the vertex set with precisely one element from each part. According to a fundamental theorem of extremal hypergraph theory, due to Erdős [7], the number of hyperedges of a d-uniform hypergraph on n vertices that does not co...

The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of a k-uniform t-intersecting family on n points, and describes all optimal families. We extend this theorem to several other settings: the weighted case, the case of infinitely many points, and the Hamming scheme. The weighted Ahlswede–Khachatrian theorem gives the maximal μp measure of a t-inte...

Intersection properties of sets have been widely investigated by many authors. One type of theorems proved for them has the following form [9]. Let S be an n-element set and AI. ... , AN £ S, 1 £ [1, n]. Assume that IAi I1Ajl E 1 for 1,,;;;; i <j ,,;;;;N. How large can N be under this condition, depending on n and I? Thus, e.g., the de Bruijn-Erdos theorem [1] asserts that if IAi I1Ajl = 1 for ...

The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of a k-uniform t-intersecting family on n points, and describes all optimal families. In recent work, we extended this theorem to the weighted setting, giving the maximum μp measure of a t-intersecting family on n points. In this work, we prove two new complete intersection theorems. The first gi...

As with many min-max characterizations, proving one of the inequalities is straightforward. For any U ⊆ S and I ∈ I1 ∩ I2, we have |I| = |I ∩ U |+ |I ∩ (S \ U)| ≤ r1(U) + r2(S \ U), since I ∩ U is an independent set in I1 and I ∩ (S \ U) is an independent set in I2. Therefore, maxI∈I1∩I2 |I| ≤ minU∈S (r1(U) + r2(S \ U)). The following important examples illustrate some of the applications of th...

We present a type system (that we refer to as System S) that features sequences as intersection types, in contrast with Gardner/de Carvalho’s System R, that uses multisets to represent intersection. Systems S and R are both linear and forbid contraction/duplication. We explain why any reduction choice in System R (i.e. any way to perform Subject Reduction) may be encoded in System S by resortin...

The main purpose of this paper is to establish a new variant the Hildebrandt?s theorem for Weyl spectrum in separable Banach space. This asserts that convex hull an operator T equal intersection numerical spectra operators are similar T.

By a knot we mean a pair (S, M~) with M~ a smooth closed oriented submanifold of S (m^3). If such a knot is given and i:S-> gm+z j s t j ^ standard embedding, then one can isotope / in an essentially unique way (Lemma 1 below) to an embedding j :s -+S whose intersection with iS is M c S transversally. The «-fold cyclic branched cover of (S, iS) branched along (jS, M~) exists uniquely and is a m...