Related papers: Set families with a forbidden subposet

For a given number of $k$-sets, how should we choose them so as to minimize the union-closed family that they generate? Our main aim in this paper is to show that, if $\mathcal{A}$ is a family of $k$-sets of size $\binom{t}{k}$, and $t$ is…

Let $n$ be any positive integer and $\mathcal{F}$ be a family of subsets of $[n]$. A family $\mathcal{F}'$ is said to be $D$-\emph{secting} for $\mathcal{F}$ if for every $A \in \mathcal{F}$, there exists a subset $A' \in \mathcal{F}'$ such…

A family $\cF \subseteq 2^{[n]}$ saturates the monotone decreasing property $\cP$ if $\cF$ satisfies $\cP$ and one cannot add any set to $\cF$ such that property $\cP$ is still satisfied by the resulting family. We address the problem of…

A family of subsets of $\{1,\ldots,n\}$ is called {\it intersecting} if any two of its sets intersect. A classical result in extremal combinatorics due to Erd\H{o}s, Ko, and Rado determines the maximum size of an intersecting family of…

A pattern class is a set of permutations closed under the formation of subpermutations. Such classes can be characterised as those permutations not involving a particular set of forbidden permutations. A simple collection of necessary and…

We say that a family of $k$-subsets of an $n$-element set is intersecting if any two of its sets intersect. In this paper we study properties and structure of large intersecting families. We prove a conclusive version of Frankl's theorem on…

We mainly investigate model of set theory with restricted choice, e.g., ZF + DC + "the family of countable subsets of lambda is well ordered for every lambda" (really local version for a given lambda). In this frame much of pcf theory can…

Let $S$ be a finite set, and $X_1,\ldots,X_n$ an i.i.d. uniform sample from $S$. To estimate the size $|S|$, without further structure, one can wait for repeats and use the birthday problem. This requires a sample size of the order…

Given any dimension function $h$, we construct a perfect set $E \subseteq \mathbb{R}$ of zero $h$-Hausdorff measure, that contains any finite polynomial pattern. This is achieved as a special case of a more general construction in which we…

A family of $k$-element subsets of an $n$-element set is called 3-wise intersecting if any three members in the family have non-empty intersection. We determine the maximum size of such families exactly or asymptotically. One of our results…

The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of $(n+1)$-element chains in the power…

For a property $\Gamma$ and a family of sets $\cF$, let $f(\cF,\Gamma)$ be the size of the largest subfamily of $\cF$ having property $\Gamma$. For a positive integer $m$, let $f(m,\Gamma)$ be the minimum of $f(\cF,\Gamma)$ over all…

When $H$ is a forest, the Gy\'arf\'as-Sumner conjecture implies that every graph $G$ with no induced subgraph isomorphic to $H$ and with bounded clique number has a stable set of linear size. We cannot prove that, but we prove that every…

Let $\mathcal{A}_1,\ldots,\mathcal{A}_m$ be families of $k$-subsets of an $n$-set. Suppose that one cannot choose pairwise disjoint edges from $s+1$ distinct families. Subject to this condition we investigate the maximum of…

Suppose that f : F_p^n -> [0,1] has expected value t in [p^(-n/9),1] (so, the density t can be quite low!). Furthermore, suppose that support(f) has no three-term arithmetic progressions. Then, we develop non-trivial lower bounds for f_j,…

A family $\mathcal C$ of sets is hereditary if whenever $A\in \mathcal C$ and $B\subset A$, we have $B\in \mathcal C$. Chv\'atal conjectured that the largest intersecting subfamily of a hereditary family is the family of all sets containing…

We provide upper bounds on the largest subsets of $\{1,2,\dots,N\}$ with no differences of the form $h_1(n_1)+\cdots+h_{\ell}(n_{\ell})$ with $n_i\in \mathbb{N}$ or $h_1(p_1)+\cdots+h_{\ell}(p_{\ell})$ with $p_i$ prime, where $h_i\in…

Let $S$ be a family of sequences of positive numbers that decrease to 0, let $X$ be a metric space and $A \subset X$. $A$ is said to be $S$-dominated if, for every $s\in S$, a countable cover $\{E_n\}$ of $E$ can be found such that $diam…

The problem of selecting a small, yet high quality subset of patterns from a larger collection of itemsets has recently attracted lot of research. Here we discuss an approach to this problem using the notion of decomposable families of…

For a given combinatorial class $\mathcal{C}$ we study the class $\mathcal{G} = \mathrm{MSET}(\mathcal{C})$ satisfying the multiset construction, that is, any object in $\mathcal{G}$ is uniquely determined by a set of $\mathcal{C}$-objects…