Related papers: Forbidden subposet problems with size restrictions
In this paper we introduce a problem that bridges forbidden subposet and forbidden subconfiguration problems. The sets $F_1,F_2, \dots,F_{|P|}$ form a copy of a poset $P$, if there exists a bijection $i:P\rightarrow \{F_1,F_2,…
Resolving a conjecture of Methuku and the first author we determine the size of the largest family of subsets of an $n$-element set avoiding both $Y_k$ and $Y_k'$ as induced subposets. The result follows as a consequence of the analogous…
We consider questions related to the computation of the capacity of codes that avoid forbidden difference patterns. The maximal number of $n$-bit sequences whose pairwise differences do not contain some given forbidden difference patterns…
We study the maximum size of a subset of the $n \times n$ integer grid that does not contain specific geometric configurations, a variation of the classical problems initiated by Erd\H{o}s and Purdy. While extremal problems for 3-point…
We discuss a possible characterization, by means of forbidden configurations, of posets which are embeddable in a product of finitely many scattered chains.
A family $\mbox{$\cal F$}=\{F_1,\ldots,F_m\}$ of subsets of $[n]$ is said to be ordered, if there exists an $1\leq r\leq m$ index such that $n\in F_i$ for each $1\leq i\leq r$, $n\notin F_i$ for each $i>r$ and $|F_i|\leq |F_j|$ for each…
In this paper, we derive a tight upper bound for the size of an intersecting $k$-Sperner family of subspaces of the $n$-dimensional vector space $\mathbb{F}_{q}^{n}$ over finite field $\mathbb{F}_{q}$ which gives a $q$-analogue of the…
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…
In the area of forbidden subposet problems we look for the largest possible size $La(n,P)$ of a family $\mathcal{F}\subseteq 2^{[n]}$ that does not contain a forbidden inclusion pattern described by $P$. The main conjecture of the area…
A family of subsets of the set {1,2,...,n} is said to be unbalanced if the convex hull of its characteristic vectors misses the diagonal in the n-cube.The purpose of this article is to develop the combinatorics of maximal unbalanced…
We introduce a new combinatorial structure: the superselector. We show that superselectors subsume several important combinatorial structures used in the past few years to solve problems in group testing, compressed sensing, multi-channel…
We consider the problem of determining the maximum number of pairs $F\subseteq F'$ in a family $\mathcal{F}\subseteq 2^{[n]}$ that avoids certain posets $P$ of height 2. We show that for any such $P$ the number of pairs is…
In this paper we relate t-designs to a forbidden configuration problem in extremal set theory. Let 1_t 0_l denote a column of t 1's on top of l 0's. We assume t>l. Let q. (1_t 0_l) denote the (t+l)xq matrix consisting of t rows of q 1's and…
In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found,…
The problem of bounding the size of a set system under various intersection restrictions has a central place in extremal combinatorics. We investigate the maximum number of disjoint pairs a set system can have in this setting. In…
Given a finite set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can we find? We make progress on three instances of subset selection problems in planar point…
We asymptotically determine the size of the largest family F of subsets of {1,...,n} not containing a given poset P if the Hasse diagram of P is a tree. This is a qualitative generalization of several known results including Sperner's…
We develop a new approach to approximate families of sets, complementing the existing `$\Delta$-system method' and `junta approximations method'. The approach, which we refer to as `spread approximations method', is based on the notion of…
We prove that for every poset $P$, there is a constant $C$ such that the size of any family of subsets of $[n]$ that does not contain $P$ as an induced subposet is at most $C{\binom{n}{\lfloor\frac{n}{2}\rfloor}}$, settling a conjecture of…
We explore from several perspectives the following question: given $X\subseteq \mathbb{Z}$ and $N\in \mathbb{N}$, what is the maximum size $D(X,N)$ of $A\subseteq \{1,2,\dots,N\}$ before $A$ is forced to contain two distinct elements that…