Induced and non-induced forbidden subposet problems

The problem of determining the maximum size $La(n,P)$ that a $P$-free subposet of the Boolean lattice $B_n$ can have, attracted the attention of many researchers, but little is known about the induced version of these problems. In this paper we determine the asymptotic behavior of $La^*(n,P)$, the maximum size that an induced $P$-free subposet of the Boolean lattice $B_n$ can have for the case when $P$ is the complete two-level poset $K_{r,t}$ or the complete multi-level poset $K_{r,s_1,\dots,s_j,t}$ when all $s_i$'s either equal 4 or are large enough and satisfy an extra condition. We also show lower and upper bounds for the non-induced problem in the case when $P$ is the complete three-level poset $K_{r,s,t}$. These bounds determine the asymptotics of $La(n,K_{r,s,t})$ for some values of $s$ independently of the values of $r$ and $t$

Supersaturation and stability for forbidden subposet problems

We address a supersaturation problem in the context of forbidden subposets. A family $\mathcal{F}$ of sets is said to contain the poset $P$ if there is an injection $i:P \rightarrow \mathcal{F}$ such that $p \le_P q$ implies $i(p) \subset i (q)$. The poset on four elements $a,b,c,d$ with $a,b \le c,d$ is called butterfly. The maximum size of a family $\mathcal{F} \subseteq 2^{[n]}$ that does not contain a butterfly is $\Sigma(n,2)=\binom{n}{\lfloor n/2 \rfloor}+\binom{n}{\lfloor n/2 \rfloor+1}$ as proved by De Bonis, Katona, and Swanepoel. We prove that if $\mathcal{F} \subseteq 2^{[n]}$ contains $\Sigma(n,2)+E$ sets, then it has to contain at least $(1-o(1))E(\lceil n/2 \rceil +1)\binom{\lceil n/2\rceil}{2}$ copies of the butterfly provided $E\le 2^{n^{1-\varepsilon}}$ for some positive $\varepsilon$. We show by a construction that this is asymptotically tight and for small values of $E$ we show that the minimum number of butterflies contained in $\mathcal{F}$ is exactly $E(\lceil n/2 \rceil +1)\binom{\lceil n/2\rceil}{2}$

On LL-close Sperner systems

For a set $L$ of positive integers, a set system $\mathcal{F} \subseteq 2^{[n]}$ is said to be $L$-close Sperner, if for any pair $F,G$ of distinct sets in $\mathcal{F}$ the skew distance $sd(F,G)=\min\{|F\setminus G|,|G\setminus F|\}$ belongs to $L$. We reprove an extremal result of Boros, Gurvich, and Milani\v c on the maximum size of $L$-close Sperner set systems for $L=\{1\}$ and generalize to $|L|=1$ and obtain slightly weaker bounds for arbitrary $L$. We also consider the problem when $L$ might include 0 and reprove a theorem of Frankl, F\"uredi, and Pach on the size of largest set systems with all skew distances belonging to $L=\{0,1\}$

On the sizes of t-intersecting k-chain-free families

A set system ${\mathcal F}$ is $t$-intersecting, if the size of the intersection of every pair of its elements has size at least $t$. A set system ${\mathcal F}$ is $k$-Sperner, if it does not contain a chain of length $k+1$. Our main result is the following: Suppose that $k$ and $t$ are fixed positive integers, where $n+t$ is even and $n$ is large enough. If ${\mathcal F}\subseteq 2^{[n]}$ is a $t$-intersecting $k$-Sperner family, then ${\mathcal F}$ has size at most the size of the sum of $k$ layers, of sizes $(n+t)/2,\ldots, (n+t)/2+k-1$. This bound is best possible. The case when $n+t$ is odd remains open

On The Ratio Of Maximum And Minimum Degree In Maximal Intersecting Families

To study how balanced or unbalanced a maximal intersecting family F subset of ((vertical bar n vertical bar) (r)) is we consider the ratio R(F) = Delta(F)/delta(F) of its maximum and minimum degree. We determine the order of magnitude of the function m(n, r), the minimum possible value of R(F), and establish some lower and upper bounds on the function M(n, r), the maximum possible value of R(F). To obtain constructions that show the bounds on m(n, r) we use a theorem of Blokhuis on the minimum size of a non-trivial blocking set in projective planes. (C) 2012 Elsevier B.V. All rights reserved.WoSScopu

Stability results for vertex Turan problems in Kneser graphs

The vertex set of the Kneser graph K(n, k) is V = (([n])(k)) and two vertices are adjacent if the corresponding sets are disjoint. For any graph F, the largest size of a vertex set U subset of V such that K(n, k)[U] is F-free, was recently determined by Alishahi and Taherkhani, whenever n is large enough compared to k and F. In this paper, we determine the second largest size of a vertex set W subset of V such that K(n, k)[W] is F-free, in the case when F is an even cycle or a complete multi-partite graph. In the latter case, we actually give a more general theorem depending on the chromatic number of F