Q2Q_2-free families in the Boolean lattice

For a family $\mathcal{F}$ of subsets of [n]=\{1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that $\mathcal{F}$ is P-free if it does not contain a subposet isomorphic to P. Let $ex(n, P)$ be the largest size of a P-free family of subsets of [n]. Let $Q_2$ be the poset with distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean lattice. We show that $2N -o(N) \leq ex(n, Q_2)\leq 2.283261N +o(N),$ where $N = \binom{n}{\lfloor n/2 \rfloor}$. We also prove that the largest $Q_2$-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.Comment: 18 pages, 2 figure

Rainbow Ramsey problems for the Boolean lattice

We address the following rainbow Ramsey problem: For posets $P,Q$ what is the smallest number $n$ such that any coloring of the elements of the Boolean lattice $B_n$ either admits a monochromatic copy of $P$ or a rainbow copy of $Q$. We consider both weak and strong (non-induced and induced) versions of this problem. We also investigate related problems on (partial) $k$-colorings of $B_n$ that do not admit rainbow antichains of size $k$

An upper bound on the size of diamond-free families of sets

Let $La(n,P)$ be the maximum size of a family of subsets of $[n]=\{1,2,...,n\}$ not containing $P$ as a (weak) subposet. The diamond poset, denoted $B_{2}$, is defined on four elements $x,y,z,w$ with the relations $x<y,z$ and $y,z<w$. $La(n,P)$ has been studied for many posets; one of the major open problems is determining $La(n,B_{2})$. Studying the average number of sets from a family of subsets of $[n]$ on a maximal chain in the Boolean lattice $2^{[n]}$ has been a fruitful method. We use a partitioning of the maximal chains and introduce an induction method to show that $La(n,B_{2})\leq(2.20711+o(1))\binom{n}{\left\lfloor \frac{n}{2}\right\rfloor }$, improving on the earlier bound of $(2.25+o(1))\binom{n}{\left\lfloor \frac{n}{2}\right\rfloor }$ by Kramer, Martin and Young.Comment: Accepted by JCTA. Writing is improved based on the suggestions of referee

Generalizations of Sperner\u27s Theorem: Packing Posets, Families Forbidding Posets, and Supersaturation

Sperner\u27s Theorem is a well known theorem in extremal set theory that gives the size of the largest antichain in the poset that is the Boolean lattice. This is equivalent to finding the largest family of subsets of an $n$-set, $[n]:=\{1,2,\dots,n\}$, such that the family is constructed from pairwise unrelated copies of the single element poset. For a poset $P$, we are interested in maximizing the size of a family $\mathcal{F}$ of subsets of $[n]$, where each maximally connected component of $\mathcal{F}$ is a copy of $P$, and finding the extreme configurations that achieve this value. For instance, Sperner showed that when $P$ is one element, $\dbinom{n}{\lfloor \frac{n}{2}\rfloor}$ is the maximum number of copies of $P$ and that this is only achieved by taking subsets of a middle size. Griggs, Stahl, and Trotter have shown that when $P$ is a chain on $k$ elements, $\dfrac{1}{2^{k-1}}\dbinom{n}{\lfloor \frac{n}{2}\rfloor}$ is asymptotically the maximum number of copies of $P$. We find the extreme families for a packing of chains, answering a conjecture of Griggs, Stahl, and Trotter, as well as finding the extreme packings of certain other posets. For the general poset $P$, we prove that the maximum number of unrelated copies of $P$ is asymptotic to a constant times $\dbinom{n}{\lfloor \frac{n}{2}\rfloor}$. Moreover, the constant has the form $\dfrac{1}{c(P)}$, where $c(P)$ is the size of the smallest convex closure over all embeddings of $P$ into the Boolean lattice. Sperner\u27s Theorem has been generalized by looking for $\operatorname{La}(n,P)$, the size of a largest family of subsets of an $n$-set that does not contain a general poset $P$ in the family. We look at this generalization, exploring different techniques for finding an upper bound on $\operatorname{La}(n,P)$, where $P$ is the diamond. We also find all the families that achieve $\operatorname{La}(n,\{\mathcal{V},\Lambda\})$, the size of the largest family of subsets that do not contain either of the posets $\mathcal{V}$ or $\Lambda$. We also consider another generalization of Sperner\u27s theorem, supersaturation, where we find how many copies of $P$ are in a family of a fixed size larger than $\operatorname{La}(n,P)$. We seek families of subsets of an $n$-set of given size that contain the fewest $k$-chains. Erd\H{o}s showed that a largest $k$-chain-free family in the Boolean lattice is formed by taking all subsets of the $(k-1)$ middle sizes. Our result implies that by taking this family together with $x$ subsets of the $k$-th middle size, we obtain a family with the minimum number of $k$-chains, over all families of this size. We prove our result using the symmetric chain decomposition method of de Bruijn, van Ebbenhorst Tengbergen, and Kruyswijk (1951)