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Maximum Antichains in Random Subsets of a Finite Set
We consider the random poset (n, p) which is generated by first selecting each subset of [n]={1, …, n} with probability p and then ordering the.

The Width of Random Subsets of Boolean Lattices - IME-USP
mines the width, that is, the maximum cardinality of an antichain, in the poset ..... [ 14] Deryk Osthus, Maximum antichains in random subsets of a finite set, ...

Lecture 12: Extremal results on finite sets 1 Largest antichains
Lecture 12: Extremal results on finite sets. Instructor: Jacob Fox. 1 Largest antichains. Suppose we are given a family F of subsets of [n]. We call F an antichain, ...

LECTURE 8. EXTREMAL SET THEORY 1. Dilworth's theorem
elements. A chain decomposition of a poset P is a partition C of its ground set into ... Ci in at most one element, each maximum antichain of size k intersects Ci ... For a finite set X of size n, define 2X as the power set of X. For a natural ... as the set of all k-subsets of X. ... of random polynomials, raised the following question.

A random version of Sperner's theorem - University of Birmingham
Aug 4, 2014 ... Let P(n) denote the power set of [n], ordered by inclusion, and let P(n, ..... [9] D. Osthus, Maximum antichains in random subsets of a finite set, ...

COMBINATORICS 1. Chains, Antichains and Shadows Definition
maximal chain has n + 1 elements, and, via a clear bijection with the set of permutations ... Choose a maximal chain C uniformly at random. Then, given F .... through the map into sums over subsets, and so Sperner's Theorem gives the result immediately. Proof. .... Let A1,...,Ak and B1,...,Bk be finite sets such that Ai ∩ Bi = ∅.

Maximum-size antichains in random set-systems
Nov 12, 2015 ... We show that, for pn → ∞, the largest set in a p-random sub-family of the ... that the largest antichain in P(n), the family of all subsets of {1,...,n}, has ...... [14] D. Osthus, Maximum antichains in random subsets of a finite set, ...

Probabilistic Methods in Extremal Finite Set Theory
Extremal Finite Set Theory is one of the most rapidly developing areas in Combinatorics, which ... antichain if no set of F is contained in another. 1 .... of estimating the maximum possible cardinality of a k-independent family of subsets of an n-set. .... let Hl be a random family of n7 (not necessarily distinct) subsets H of N ...

Combinatorics of finite sets by Ian Anderson, Clarendon Press
13. Yu. L. Pavlov, The asymptotic distribution of maximum tree size in a random forest,. Theor. ... Ian Anderson's Combinatorics of finite sets brings to maturity a small but elegant area ... primary object of study here is the subset lattice, which is the inclusion or- ... In 1928, Sperner determined the maximum-sized antichains in.

Probabilistic Methods in Combinatorics
and suppose that none of the Ai are subsets of each other. For each ... Let X be a random variable which takes values in some finite set S. Then the expected value ..... (Sperner's Lemma) Prove that the maximum antichain in 2[n] has size ( n.

On scattered convex geometries
May 12, 2015 ... set. In particular, the closed sets of an algebraic convex geometry form an algebraic lattice, i.e. a ... the closures of finite subsets of X, and they form a semilattice with respect to the join ..... Indeed, pick two points x1,x2 from X randomly, and if x1 ..... an element below or above all the elements of the antichain.

The limit surface of antichains in the random 3-dimensional partial
set, called the random 3-dimensional partial order, with a focus on the rare event that it is an ... An antichain is a subset A ⊆ P such that every two elements of A ..... Theorem. The set of maximum-sized antichains in a finite partial order P.

Notes (1 November 2015)
Most of this course concerns properties of set systems with a finite ground set. So from now on all ... {A ∈ P(X) : |A| = k} for the set of all subsets of X of size k. We define. X(<k) .... Sperner's Lemma tells us the maximal size of an antichain, but what can we say about .... at random from all n! maximal chains. For A ∈ F with |A | ...

Here
Mar 10, 2010 ... will prove that both (i) the finite Boolean algebra 2[n] and (ii) L(m, n), the set of all partitions with at most m parts and largest part at most n, both .... Just one more definition: an antichain in a poset P is a subset A ⊂ P for which .... useful: the cusps on the profile of a (random) tableaux projected down to the x- ...

Ordered Sets
Nov 27, 2001 ... These notes will be updated at more or less random times during the term. 1. Consider the .... The size of the largest antichain in an ordered set is called its width; ... irreducible elements in the lattice of subsets of a finite set?

Intersecting Families of Finite Sets and Fixed-point-Free 2-Elements
Intersecting Families of Finite Sets and Fixed-point-Free 2-Elements ... We study the maximum cardinality of a pairwise-intersecting family of subsets of an n-set, or ... subsets of an n-element set X: typically, these are bounds on the cardinality of .... random partition of Xinto 10 parts, labelled with the 2-subsets of {l, ... ,5}, say  ...

Antichains: a new algorithm for checking universality of finite automata
algorithm experimentally using the random automaton model recently proposed ... us to store only maximal subsets of states for which a winning strategy exists. ... tomata can be solved on the lattice of antichains of state sets using a variation.

A new proof of the density Hales-Jewett theorem - Annals of
Jul 2, 2011 ... subset of [k]n of density at least δ, then A contains a combinatorial line. ..... Lemma 2.2. Let X be a finite set, and let Xγ be a random subset of X, .... is that the maximum possible equal-slices measure of an antichain is 1/(n+ 1).