A note on blockers in posets

The blocker $A^{*}$ of an antichain $A$ in a finite poset $P$ is the set of elements minimal with the property of having with each member of $A$ a common predecessor. The following is done: 1. The posets $P$ for which $A^{**}=A$ for all antichains are characterized. 2. The blocker $A^*$ of a symmetric antichain in the partition lattice is characterized. 3. Connections with the question of finding minimal size blocking sets for certain set families are discussed

Minimum Weight Flat Antichains of Subsets

Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains $\mathcal F$ in the Boolean lattice $B_n$ of all subsets of $[n]:=\{1,2,\dots,n\}$, where $\mathcal F$ is flat, meaning that it contains sets of at most two consecutive sizes, say $\mathcal F=\mathcal{A}\cup\mathcal{B}$, where $\mathcal{A}$ contains only $k$-subsets, while $\mathcal{B}$ contains only $(k-1)$-subsets. Moreover, we assume $\mathcal{A}$ consists of the first $m$ $k$-subsets in squashed (colexicographic) order, while $\mathcal{B}$ consists of all $(k-1)$-subsets not contained in the subsets in $\mathcal{A}$. Given reals $\alpha,\beta>0$, we say the weight of $\mathcal F$ is $\alpha\cdot|\mathcal{A}|+\beta\cdot|\mathcal{B}|$. We characterize the minimum weight antichains $\mathcal F$ for any given $n,k,\alpha,\beta$, and we do the same when in addition $\mathcal F$ is a maximal antichain. We can then derive asymptotic results on both the minimum size and the minimum Lubell function

On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders

We investigate the ordinal invariants height, length, and width of well quasi orders (WQO), with particular emphasis on width, an invariant of interest for the larger class of orders with finite antichain condition (FAC). We show that the width in the class of FAC orders is completely determined by the width in the class of WQOs, in the sense that if we know how to calculate the width of any WQO then we have a procedure to calculate the width of any given FAC order. We show how the width of WQO orders obtained via some classical constructions can sometimes be computed in a compositional way. In particular, this allows proving that every ordinal can be obtained as the width of some WQO poset. One of the difficult questions is to give a complete formula for the width of Cartesian products of WQOs. Even the width of the product of two ordinals is only known through a complex recursive formula. Although we have not given a complete answer to this question we have advanced the state of knowledge by considering some more complex special cases and in particular by calculating the width of certain products containing three factors. In the course of writing the paper we have discovered that some of the relevant literature was written on cross-purposes and some of the notions re-discovered several times. Therefore we also use the occasion to give a unified presentation of the known results

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