Kirszbraun’s Theorem states that every Lipschitz map S → Rn, where S ⊆ Rm, has an extension to a Lipschitz map Rm → Rn with the same Lipschitz constant. Its proof relies on Helly’s Theorem: every family of compact subsets of Rn, having the property that each of its subfamilies consisting of at most n + 1 sets share a common point, has a non-empty intersection. We prove versions of these theorem...

A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that so...

In this paper both blocking sets with respect to the s-subspaces and covers with t-subspaces in a finite Grassmannian are investigated, especially focusing on geometric descriptions of blocking sets and covers of minimum size. When such a description exists, it is called a Bose–Burton type theorem. The canonical example of a blocking set with respect to the s-subspaces is the intersection of s ...

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 ...

We first present new structural properties of a two-pair in various graphs. A twopair is used in a well-known characterization of weakly chordal graphs. Based on these properties, we prove the main theorem: a graph G is a weakly chordal (K2,3, 4P2, P2 ∪ P4, P6,H1,H2,H3)-free graph if and only if G is an edge intersection graph of subtrees on a tree with maximum degree 4. This characterizes the ...

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...

We study intersection cohomology complexes on flag varieties with coefficients in a field of positive characteristic and present a combinatorial procedure (based on the W -graph of the Coxeter group) which determines their characters in many cases on low rank flag varieties. Our procedure works uniformly in almost all characteristics (p > 5 is always sufficient) and, if successful, verifies a v...

In 1976 Hirzebruch and Zagier [6] computed the pairwise intersection multiplicities for a family of algebraic cycles (T£)n(EN in the Hubert modular surface associated to Q(yp), where p is a prime congruent to 1 mod 4, and showed that the generating function for those intersection multiplicities Yln°=oC^m 'T£)e[nr] was an elliptic modular form of weight 2 and Nebentypus for To(p). Shortly afterw...

Abstract We introduced the fuzzy axioms of choice, Zorn’s lemma and well-ordering principle, which are versions discussed relations among them. As an application lemma, we got following results: (1) Every proper ideal a ring was contained in maximal ideal. (2) nonzero (3) Introduced notion nilpotent elements R , proved that intersection all prime ideals commutative is union . (4) Proposed versi...