Sperner's problem for G-independent families

Given a graph G, let Q(G) denote the collection of all independent (edge-free) sets of vertices in G. We consider the problem of determining the size of a largest antichain in Q(G). When G is the edge-less graph, this problem is resolved by Sperner's Theorem. In this paper, we focus on the case where G is the path of length n-1, proving the size of a maximal antichain is of the same order as the size of a largest layer of Q(G).Comment: 26 page

Regular Intersecting Families

We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families $\mathcal{F}$ of $k$-element subsets of $[n]:=\{1,\ldots, n\},$ such that every element of $[n]$ lies in the same (or approximately the same) number of members of $\mathcal{F}$. In particular, we show that we can guarantee $|\mathcal{F}| = o({n-1\choose k-1})$ if and only if $k=o(n)$.Comment: 15 pages, accepted versio

Probabilistic Inequalities and Bootstrap Percolation

This dissertation focuses on two topics. In the first part of we address a number of extremal probabilistic questions:The Littlewood-Offord problem: we provide an alternative and very elementary proof of a classical result by Erdos that avoids using Sperner\u27s Theorem. We also give a new simple proof of Sperner\u27s Theorem itself.Upper bounds for the concentration function: answering a question of Leader and Radcliffe we obtain optimal upper bounds for the concentration function of a sum of real random variables when individual concentration information about the summands is given. The result can be viewed as the optimal form of a well-known Kolmogorov-Rogozin inequality.Small ball probabilities for sums of random vectors with bounded density: we provide optimal upper bounds the probability that a sum of random vectors lies inside a small ball and derive an upper bound for the maximum density of this sum. In particular, our work extends a result of Rogozin who proved the best possible result in one dimension and improves some recent results proved by Bobkov and Chystiakov.Two extremal questions of bounded symmetric random walks: we obtain optimal upper bounds for the probability that a sum of independent bounded symmetric random variables exceeds a given value or hits it.The second part of the dissertation is concerned with a problem in Bootstrap Percolation. Let G be a graph and let I be a set of initially infected vertices. The set of infected vertices is updated as follows: if a healthy vertex has the majority of its neighbours infected it itself becomes infected.In the description of the bootstrap process above the superscripts of the sets correspond to the time steps when infections occur. If the process ends up infecting all of the vertices we say that percolation occurs.In this dissertation we shall investigate this process on the Erdos-Renyi random graph G(n,p). In this graph on n vertices each edge is included independently with probability p. We shall be interested in the smallest cardinality, say m=m(n), of a uniformly chosen initially infected set of vertices I, such that the probability of percolation at least 1/2. We call this quantity the critical size of the initially infected set. In the regime p\u3eclog (n)/n (the connectivity threshhold) we prove sharp upper and lower bounds for m that match in the first two terms of the asymptotic expansion

Dually conformal hypergraphs

Given a hypergraph $\mathcal{H}$, the dual hypergraph of $\mathcal{H}$ is the hypergraph of all minimal transversals of $\mathcal{H}$. The dual hypergraph is always Sperner, that is, no hyperedge contains another. A special case of Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to the families of maximal cliques of graphs. All these notions play an important role in many fields of mathematics and computer science, including combinatorics, algebra, database theory, etc. In this paper we study conformality of dual hypergraphs. While we do not settle the computational complexity status of recognizing this property, we show that the problem is in co-NP and can be solved in polynomial time for hypergraphs of bounded dimension. In the special case of dimension $3$, we reduce the problem to $2$-Satisfiability. Our approach has an implication in algorithmic graph theory: we obtain a polynomial-time algorithm for recognizing graphs in which all minimal transversals of maximal cliques have size at most $k$, for any fixed $k$