Kunal Dutta

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We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let V be a finite set system defined over an n-point set X; we view V as a set of indicator vectors over the n-dimensional unit cube. A δ-separated set of V is a subcollection W, s.t. the Hamming distance between each pair u,v ∈ W is(More)
We obtain new lower bounds for the independence number of Kr-free graphs and linear kuniform hypergraphs in terms of the degree sequence. This answers some old questions raised by Caro and Tuza [7]. Our proof technique is an extension of a method of Caro and Wei [6, 20], and we also give a new short proof of the main result of [7] using this approach. As(More)
The packing lemma of Haussler states that given a set system (X,R) with bounded VC dimension, if every pair of sets in R have large symmetric difference, then R cannot contain too many sets. Recently it was generalized to the shallow packing lemma, applying to set systems as a function of their shallow-cell complexity. In this paper we present several new(More)
A subset D ⊆ V of a graph G = (V,E) is a (1, j)-set if every vertex v ∈ V \D is adjacent to at least 1 but not more than j vertices in D. The cardinality of a minimum (1, j)-set of G, denoted as γ(1,j)(G), is called the (1, j)-domination number of G. Given a graph G = (V,E) and an integer k, the decision version of the (1, j)-set problem is to decide(More)
Let G be a triangle-free graph with n vertices and average degree t. We show that G contains at least e(1−n −1/12) 1 2 n t ln t( 1 2 ln t−1) independent sets. This improves a recent result of the first and third authors [8]. In particular, it implies that as n → ∞, every triangle-free graph on n vertices has at least e(c1−o(1)) √ n lnn independent sets,(More)