Kunal Dutta

New Lower Bounds for the Independence Number of Sparse Graphs and Hypergraphs

We obtain new lower bounds for the independence number of

Combinatorics of finite abelian groups and Weil representations

K_r

A simple proof of optimal epsilon nets

-free graphs and linear

Counting Independent Sets in Hypergraphs

k

Degenerations and orbits in finite abelian groups

-uniform hypergraphs in terms of the degree sequence. This answers some old questions raised by Caro and Tu...

Degeneration and orbits of tuples and subgroups in an Abelian group

The Weil representation of the symplectic group associated to a finite abelian group of odd order is shown to have a multiplicity-free decom- position. When the abelian group is p-primary, the irreducible representations occurring in the Weil representation are parametrized by a partially ordered set which is independent of p. As p varies, the dimension of the irreducible rep- resentation corresponding to each parameter is shown to be a polynomial in p which is calculated explicitly. The commuting algebra of the Weil representa- tion has a basis indexed by another partially ordered set which is independent of p. The expansions of the projection operators onto the irreducible invari- ant subspaces in terms of this basis are calculated. The coefficients are again polynomials in p. These results remain valid in the more general setting of finitely generated torsion modules over a Dedekind domain.

Kernelization of the subset general position problem in geometry

Showing the existence of small-sized epsilon-nets has been the subject of investigation for almost 30 years, starting from the breakthrough of Haussler and Welzl (1987). Following a long line of successive improvements, recent results have settled the question of the size of the smallest epsilon-nets for set systems as a function of their so-called shallow cell complexity. In this paper we give a short proof of this theorem in the space of a few elementary paragraphs. This immediately implies all known cases of results on unweighted epsilon-nets studied for the past 28 years, starting from the result of Matou\v{s}ek, Seidel and Welzl (SoCG 1990) to that of Clarkson and Varadarajan (DCG 2007) to that of Varadarajan (STOC 2010) and Chan et al. (SODA 2012) for the unweighted case, as well as the technical and intricate paper of Aronov et al. (SIAM Journal on Computing, 2010). We find it quite surprising that such a simple elementary approach was missed in all earlier work, as all ingredients were already known in 1991.

Improved Bounds on Induced Acyclic Subgraphs in Random Digraphs

Let

( 1 , j ) -set problem in graphs

G

Largest induced acyclic tournament in random digraphs: a 2-point concentration

be a triangle-free graph with