A random version of Sperner's theorem

Let $\mathcal{P}(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\mathcal{P}(n,p)$ be obtained from $\mathcal{P}(n)$ by selecting elements from $\mathcal{P}(n)$ independently at random with probability $p$. A classical result of Sperner asserts that every antichain in $\mathcal{P}(n)$ has size at most that of the middle layer, $\binom{n}{\lfloor n/2 \rfloor}$. In this note we prove an analogous result for $\mathcal{P} (n,p)$: If $pn \rightarrow \infty$ then, with high probability, the size of the largest antichain in $\mathcal{P}(n,p)$ is at most $(1+o(1)) p \binom{n}{\lfloor n/2 \rfloor}$. This solves a conjecture of Osthus who proved the result in the case when $pn/\log n \rightarrow \infty$. Our condition on $p$ is best-possible. In fact, we prove a more general result giving an upper bound on the size of the largest antichain for a wider range of values of $p$.Comment: 7 pages. Updated to include minor revisions and publication dat

Improved bounds for cross-Sperner systems

A collection of families (F1,F2,⋯,Fk)∈P([n])k is cross-Sperner if there is no pair i≠j for which some Fi∈Fi is comparable to some Fj∈Fj. Two natural measures of the 'size' of such a family are the sum ∑ki=1|Fi| and the product ∏ki=1|Fi|. We prove new upper and lower bounds on both of these measures for general n and k≥2 which improve considerably on the previous best bounds. In particular, we construct a rich family of counterexamples to a conjecture of Gerbner, Lemons, Palmer, Patkós, and Szécsi from 2011

On some extremal and probabilistic questions for tree posets

Given two posets $P,Q$ we say that $Q$ is $P$-free if $Q$ does not contain a copy of $P$. The size of the largest $P$-free family in $2^{[n]}$, denoted by $La(n,P)$, has been extensively studied since the 1980s. We consider several related problems. Indeed, for posets $P$ whose Hasse diagrams are trees and have radius at most $2$, we prove that there are $2^{(1+o(1))La(n,P)}$ $P$-free families in $2^{[n]}$, thereby confirming a conjecture of Gerbner, Nagy, Patk\'os and Vizer [Electronic Journal of Combinatorics, 2021] in these cases. For such $P$ we also resolve the random version of the $P$-free problem, thus generalising the random version of Sperner's theorem due to Balogh, Mycroft and Treglown [Journal of Combinatorial Theory Series A, 2014], and Collares Neto and Morris [Random Structures and Algorithms, 2016]. Additionally, we make a general conjecture that, roughly speaking, asserts that subfamilies of $2^{[n]}$ of size sufficiently above $La(n,P)$ robustly contain $P$, for any poset $P$ whose Hasse diagram is a tree

Random and deterministic versions of extremal poset problems

Let P(n) denote the set of all subsets of {1,...,n} and let P(n,p) be the set obtained from P(n) by selecting elements independently at random with probability p. The Boolean lattice is a partially ordered set, or poset, consisting of the elements of P(n), partially ordered by set inclusion. A basic question in extremal poset theory asks the following: Given a poset P, how big is the largest family of sets in the Boolean lattice which does not contain the structure P as a subposet? The following random analogue of this question is also of interest: Given a poset P, how big is the largest family of sets in P(n,p) which does not contain the structure P as a subposet? In this thesis, we present new proofs for a collection of deterministic extremal subposet problems. We also discuss a new technique called the Hypergraph Container Method in depth and use it to prove a random version of De Bonis and Katona\u27s (r+1)-fork-free theorem

Existence thresholds and Ramsey properties of random posets

Let $\mathcal P(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\mathcal P (n,p)$ denote the random poset obtained from $\mathcal P(n)$ by retaining each element from $\mathcal P (n)$ independently at random with probability $p$ and discarding it otherwise. Given any fixed poset $F$ we determine the threshold for the property that $\mathcal P(n,p)$ contains $F$ as an induced subposet. We also asymptotically determine the number of copies of a fixed poset $F$ in $\mathcal P(n)$. Finally, we obtain a number of results on the Ramsey properties of the random poset $\mathcal P(n,p)$.Comment: 33 pages, 2 figures. Author accepted manuscript, to appear in Random Structures and Algorithm

Maximum antichains in random subsets of a finite set

AbstractWe consider the random poset P(n,p) which is generated by first selecting each subset of [n]={1,…,n} with probability p and then ordering the selected subsets by inclusion. We give asymptotic estimates of the size of the maximum antichain for arbitrary p=p(n). In particular, we prove that if pn/logn→∞, an analogue of Sperner's theorem holds: almost surely the maximum antichain is (to first order) no larger than the antichain which is obtained by selecting all elements of P(n,p) with cardinality ⌊n/2⌋. This is almost surely not the case if pn=↛∞