An $n$-vertex graph $G$ of edge density $p$ is considered to be quasirandom if it shares several important properties with the random graph $G(n,p)$. A well-known theorem of Chung, Graham and Wilson states that many such `typical' properties are asymptotically equivalent and, thus, a graph $G$ possessing one such property automatically satisfies the others.In recent years, work in this area has focused on uncovering more quasirandom graph properties and on extending the known results to other discrete structures. In the context of hypergraphs, however, one may consider several different notions of quasirandomness. A complete description of these notions has been provided recently by Towsner, who proved several central equivalences using an analytic framework. We give short and purely combinatorial proofs of the main equivalences in Towsner's result.

Abstract. An n-vertex graph G of edge density p is considered to be quasirandom if it shares several important properties with the random graph Gpn, pq. A well-known theorem of Chung, Graham and Wilson states that many such 'typical' properties are asymptotically equivalent and, thus, a graph G possessing one such property automatically satisfies the others.In recent years, work in this area has focused on uncovering more quasirandom graph properties and on extending the known results to other discrete structures. In the context of hypergraphs, however, one may consider several different notions of quasirandomness. A complete description of these notions has been provided recently by Towsner, who proved several central equivalences using an analytic framework. We give short and purely combinatorial proofs of the main equivalences of Towsner's result. §1. IntroductionQuasirandomness may be seen as the study of structures which share some of the typical properties of a random structure of the same size. This area has connections to and applications in several branches of pure mathematics and theoretical computer science. For further information, we refer the reader to the surveys [22,23,39]. We focus here on quasirandom graphs and hypergraphs.Let pG n q nPN be a sequence of graphs, where G n is a graph on n vertices. For a fixed p P r0, 1s, we say that pG n q nPN is p-quasirandom if the graphs G n have a uniform edge distribution and density p, that is,where epG n rSsq denotes the number of edges in the induced subgraph G n rSs. property in the sense that a sequence pG n q nPN satisfying property (1.1) will also satisfy several other properties typically expected (with high probability) of the random graph Gpn, pq. For

No abstract

Abstract. In FOCS 2002, Even et al. showed that any set of n discs in the plane can be Conflict-Free colored with a total of at most O(log n) colors. That is, it can be colored with O(log n) colors such that for any (covered) point p there is some disc whose color is distinct from all other colors of discs containing p. They also showed that this bound is asymptotically tight. In this paper we prove the following stronger results:(i) Any set of n discs in the plane can be colored with a total of at most O(k log n) colors such that (a) for any point p that is covered by at least k discs, there are at least k distinct discs each of which is colored by a color distinct from all other discs containing p and (b) for any point p covered by at most k discs, all discs covering p are colored distinctively. We call such a coloring a k-Strong Conflict-Free coloring. We extend this result to pseudo-discs and arbitrary regions with linear union-complexity. (ii) More generally, for families of n simple closed Jordan regions with union-complexity bounded by O(n 1+α ), we prove that there exists a k-Strong Conflict-Free coloring with at most O(kn α ) colors. (iii) We prove that any set of n axis-parallel rectangles can be k-Strong Conflict-Free colored with at most O(k log 2 n) colors. (iv) We provide a general framework for k-Strong Conflict-Free coloring arbitrary hypergraphs. This framework relates the notion of k-Strong Conflict-Free coloring and the recently studied notion of k-colorful coloring. All of our proofs are constructive. That is, there exist polynomial time algorithms for computing such colorings.

We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved recently by Aharoni and Berger.We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids.In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays. * Research supported by the Minerva foundation. 1 see Theorem 3.1 below.