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Harremoes P.,Copenhagen Business School | Tusnady G.,Renyi Institute of Mathematics
IEEE International Symposium on Information Theory - Proceedings | Year: 2012

For testing goodness of fit it is very popular to use either the χ 2-statistic or G 2-statistics (information divergence). Asymptotically both are χ 2-distributed so an obvious question is which of the two statistics that has a distribution that is closest to the χ 2-distribution. Surprisingly, when there is only one degree of freedom it seems like the distribution of information divergence is much better approximated by a χ 2-distribution than the χ 2-statistic. For random variables we introduce a new transformation that transform several important distributions into new random variables that are almost Gaussian. For the binomial distributions and the Poisson distributions we formulate a general conjecture about how close their transform are to the Gaussian. The conjecture is proved for Poisson distributions. © 2012 IEEE.

Furedi Z.,Renyi Institute of Mathematics | Jiang T.,Miami University Ohio
Journal of Combinatorial Theory. Series A | Year: 2014

A k-uniform linear cycle of length ℓ, denoted by Cℓ(k), is a cyclic list of k-sets A1, . . . , Aℓ such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint. For all k ≥ 5 and ℓ ≥ 3 and sufficiently large n we determine the largest size of a k-uniform set family on [n] not containing a linear cycle of length ℓ. For odd ℓ = 2t + 1 the unique extremal family FS consists of all k-sets in [n] intersecting a fixed t-set S in [n]. For even ℓ = 2t + 2, the unique extremal family consists of FS plus all the k-sets outside S containing some fixed two elements. For k ≥ 4 and large n we also establish an exact result for so-called minimal cycles. For all k ≥ 4 our results substantially extend Erdos's result on largest k-uniform families without t + 1 pairwise disjoint members and confirm, in a stronger form, a conjecture of Mubayi and Verstraëte. Our main method is the delta system method. © 2014 Elsevier Inc.

Pach J.,Ecole Polytechnique Federale de Lausanne | Tardos G.,Renyi Institute of Mathematics
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2014

Given a sequence of positive integers p = (p1,…, pn), let Sp denote the family of all sequences of positive integers x = (x1,…, xn) such that xi ≤ pi for all i. Two families of sequences (or vectors), A, B ⊆ Sp, are said to be r-cross-intersecting if no matter how we select x ∈ A and y ∈ B, there are at least r distinct indices i such that xi = yi. We determine the maximum value of |A| · |B| over all pairs of r-cross-intersecting families and characterize the extremal pairs for r ≥ 1, provided that min pi > r + 1. The case min pi ≤ r + 1 is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Frankl, Füredi, Livingston, Moon, and Tokushige, and answers a question of Zhang. The special case r = 1 has also been settled recently by Borg. © Springer International Publishing Switzerland 2014.

Pach J.,Ecole Polytechnique Federale de Lausanne | Radoicic R.,City University of New York | Toth G.,Renyi Institute of Mathematics
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2012

A tangle is a graph drawn in the plane so that any pair of edges have precisely one point in common, and this point is either an endpoint or a point of tangency. If we allow a third option: the common point may be a proper crossing between the two edges, then the graph is called a tangled thrackle. We establish the following analogues of Conway's thrackle conjecture: The number of edges of a tangle cannot exceed its number of vertices, n. We also prove that the number of edges of an x-monotone tangled thrackle with n vertices is at most n + 1. Both results are tight for n > 3. For not necessarily x-monotone tangled thrackles, we have a somewhat weaker, but nearly linear, upper bound. © 2012 Springer-Verlag.

Csirmaz L.,Central European University | Tardos G.,Simon Fraser University | Tardos G.,Renyi Institute of Mathematics
IEEE Transactions on Information Theory | Year: 2013

The information rate for an access structure is the reciprocal of the load of the optimal secret sharing scheme for this structure. We determine this value for all trees: it is , where is the size of the largest core of the tree. A subset of the vertices of a tree is a core if it induces a connected subgraph and for each vertex in the subset one finds a neighbor outside the subset. Our result follows from a lower and an upper bound on the information rate that applies for any graph and happen to coincide for trees because of a correspondence between the size of the largest core and a quantity related to a fractional cover of the tree with stars.© 2012 IEEE.

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