For a graph G, let ff(G) denote the size of the largest independent set in G, and let #(G) denote the Lov'asz #-function on G. We prove that for some c ? 0, there exists an infinite family of graphs such that #(G) ? ff(G)n=2 c p log n , where n denotes the number of vertices in a graph. This disproves a known conjecture regarding the # function. As part of our proof, we analyse the behavior of the chromatic number in graphs under a randomized version of graph products. This analysis extends earlier work of Linial and Vazirani, and of Berman and Schnitger, and may be of independent interest. 1 Introduction Lov'asz [21] introduced the # function in order to study the so called "Shannon Capacity" of graphs. For every graph G, the # function enjoys the following sandwich property: ff(G) #(G) (G) where ff(G) is the size of the largest independent set in G, and (G) is the clique cover number of G ((G) = ( G), the chromatic number of the complement of G). This sandwich prop...

The PAC learning of rectangles has been studied because they have been found experimentally to yield excellent hypotheses for several applied learning problems. Also, pseudorandom sets for rectangles have been actively studied recently because (i) they are a subproblem common to the derandomization of depth-2 (DNF) circuits and derandomizing Randomized Logspace, and (ii) they approximate the distribution of n independent multivalued random variables. We present improved upper bounds for a class of such problems of "approximating" high-dimensional rectangles that arise in PAC learning and pseudorandomness. Key words and phrases. Rectangles, machine learning, PAC learning, derandomization, pseudorandomness, multiple-instance learning, explicit constructions, Ramsey graphs, random graphs, sample complexity, approximations of distributions. 2 1 Introduction A basic common theme of a large part of PAC learning and derandomization/computational pseudorandomness is to "approximate" a stru...

We present new explicit constructions of deterministic randomness extractors, dispersers and related objects. More precisely, a distribution X over binary strings of length n is called a δ-source if it assigns probability at most 2 −δn to any string of length n, and for any δ> 0 we construct the following poly(n)-time computable functions: 2-source disperser: D: ({0, 1} n) 2 → {0, 1} such that for any two independent δ-sources X1, X2 we have that the support of D(X1, X2) is {0, 1}. Bipartite Ramsey graph: Let N = 2 n. A corollary is that the function D is a 2-coloring of the edges of KN,N (the complete bipartite graph over two sets of N vertices) such that any induced subgraph of size N δ by N δ is not monochromatic. 3-source extractor: E: ({0, 1} n) 2 → {0, 1} such that for any three independent δ-sources X1, X2, X3 we have that E(X1, X2, X3) is (o(1)-close to being) an unbiased random bit. No previous explicit construction was known for either of these, for any δ < 1/2 and these results constitute major progress to long-standing open problems. A component in these results is a new construction of condensers that may be of independent

Abstract. We give the first exponential separation between quantum and bounded-error randomized one-way communication complexity. Specifically, we define the Hidden Matching Problem HMn: Alice gets as input a string x ∈ {0, 1} n and Bob gets a perfect matching M on the n coordinates. Bob’s goal is to output a tuple 〈i, j, b 〉 such that the edge (i, j) belongs to the matching M and b = xi ⊕ xj. We prove that the quantum one-way communication complexity of HMn is O(log n), yet any randomized one-way protocol with bounded error must use Ω ( √ n) bits of communication. No asymptotic gap for one-way communication was previously known. Our bounds also hold in the model of Simultaneous Messages (SM) and hence we provide the first exponential separation between quantum SM and randomized SM with public coins. For a Boolean decision version of HMn, we show that the quantum one-way communication complexity remains O(log n) and that the 0-error randomized one-way communication complexity is Ω(n). We prove that any randomized linear one-way protocol with bounded error for this problem requires Ω ( 3 √ n log n) bits of communication. Key words. Communication complexity, quantum computation, separation, hidden matching AMS subject classifications. 68P30,68Q15,68Q17,81P68 1. Introduction. The

We construct a system H of exp(c log 2 n= log log n) subsets of a set of n elements such that the size of each set is divisible by 6 but their pairwise intersections are not divisible by 6. The result generalizes to all non-prime-power moduli m in place of m = 6. This result is in sharp contrast with results of Frankl and Wilson (1981) for prime power moduli and gives strong negative answers to questions by Frankl and Wilson (1981) and Babai and Frankl (1992). We use our set-system H to give an explicit Ramsey-graph construction, reproducing the logarithmic order of magnitude of the best previously known construction due to Frankl and Wilson (1981). Our construction uses certain mod m polynomials, discovered by Barrington, Beigel and Rudich (1994). 1 Introduction Generalizing the Ray-Chaudhuri|Wilson theorem [8], Frankl and Wilson [6] proved the following intersection theorem, one of the most important results in extremal set theory: Department of Computer Science, Eotvos Un...

For every xed integers r; s satisfying 2 r < s there exists some = (r; s) > 0 for which we construct explicitly an innite family of graphs H r;s;n , where H r;s;n has n vertices, contains no clique on s vertices and every subset of at least n 1 of its vertices contains a clique of size r. The constructions are based on spectral and geometric techniques, some properties of Finite Geometries and certain isoperimetric inequalities. 1

For an undirected graph G = (V; E), let G n denote the graph whose vertex set is V n in which two distinct vertices (u 1 ; u 2 ; : : : ; un ) and (v 1 ; v 2 ; : : : ; v n ) are adjacent iff for all i between 1 and n either u i = v i or u i v i 2 E. The Shannon capacity c(G) of G is the limit limn7!1 (ff(G n )) 1=n , where ff(G n ) is the maximum size of an independent set of vertices in G n . We show that there are graphs G and H such that the Shannon capacity of their disjoint union is (much) bigger than the sum of their capacities. This disproves a conjecture of Shannon raised in 1956. 1 Introduction For an undirected graph G = (V; E), let G n denote the graph whose vertex set is V n in which two distinct vertices (u 1 ; u 2 ; : : : ; u n ) and (v 1 ; v 2 ; : : : ; v n ) are adjacent iff for all i between 1 and n either u i = v i or u i v i 2 E. The Shannon capacity c(G) of G is the limit lim n7!1 (ff(G n )) 1=n , where ff(G n ) is the maximum size of an inde...

The main result of this paper is an explicit disperser for two independent sources on n bits, each of entropy k = n o(1). Put differently, setting N = 2 n and K = 2 k, we construct explicit N × N Boolean matrices for which no K × K submatrix is monochromatic. Viewed as adjacency matrices of bipartite graphs, this gives an explicit construction of K-Ramsey bipartite graphs of size N. This greatly improves the previous bound of k = o(n) of Barak, Kindler, Shaltiel, Sudakov and Wigderson [4]. It also significantly improves the 25-year record of k = Õ( √ n) on the special case of Ramsey graphs, due to Frankl and Wilson [9]. The construction uses (besides ”classical ” extractor ideas) almost all of the machinery developed in the last couple of years for extraction from independent sources, including: • Bourgain’s extractor for 2 independent sources of some entropy rate < 1/2 [5] • Raz’s extractor for 2 independent sources, one of which has any entropy rate> 1/2 [18] • Rao’s extractor for 2 independent block-sources of entropy n Ω(1) [17]

Let f =(f1,...,fm) be a sequence of polynomials of degree ≤ d in n variables (m ≥ n) overafieldF. The zero-pattern of f at u ∈ F n is the set of those i (1 ≤ i ≤ m) forwhichfi(u) =0. LetZF (f) denote the number of zero-patterns of f as u ranges over F n.WeprovethatZF (f) ≤ �n � � m j=0 j for d =1and md

We examine n×n matrices over Zm, with 0’s in the diagonal and nonzeros elsewhere. If m is a prime, then such matrices have large rank (i.e., n 1/(p−1) − O(1)). If m is a non-prime-power integer, then we show that their rank can be much smaller. For m = 6 we construct a matrix of rank exp(c √ log n log log n). We also show, that explicit constructions of such low rank matrices imply explicit constructions of Ramsey graphs. Keywords: composite modulus, explicit Ramsey-graph constructions, matrices over rings, co-diagonal matrices 1