Related papers: Lubell mass and induced partially ordered sets

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…

A family of sets F is said to be union-closed if A \cup B is in F for every A and B in F. Frankl's conjecture states that given any finite union-closed family of sets, not all empty, there exists an element contained in at least half of the…

Let $P$ be a partially ordered set. The function $\mbox{La}^{\#}(n,P)$ denotes the size of the largest family $\mathcal{F}\subset 2^{[n]}$ that does not contain an induced copy of $P$. It was proved by Methuku and P\'alv\"olgyi that there…

We show that there is some absolute constant $c>0$, such that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}$, if \mbox{$|\mathcal{F}| \geq (\frac{1}{2}-c)2^n$}, then there is some element $i \in [n]$ that appears in at least…

A family $\mathcal{G}$ of sets is a(n induced) copy of a poset $P=(P,\leqslant)$ if there exists a bijection $b:P\rightarrow \mathcal{G}$ such that $p\leqslant q$ holds if and only if $b(p)\subseteq b(q)$. The induced saturation number…

We find previously unknown families of sets which ensure Frankl's conjecture holds for all families that contain them using an algorithmic framework. The conjecture states that for any nonempty union-closed (UC) family there exists an…

A subfamily $\mathcal{G}\subseteq \mathcal{F}\subseteq 2^{[n]}$ of sets is a non-induced (weak) copy of a poset $P$ in $\mathcal{F}$ if there exists a bijection $i:P\rightarrow \mathcal{G}$ such that $p\le_P q$ implies $i(p)\subseteq i(q)$.…

The Frankl or Union-Closed Sets conjecture states that for any finite union-closed family of sets $\mathcal{F}$ containing some nonempty set, there is some element $i$ in the ground set $U(\mathcal F) := \bigcup_{S \in \mathcal{F}} S$ of…

A family of sets is union-closed (UC) if the union of any two sets in the family is also in the family. Frankl's UC sets conjecture states that for any nonempty UC family $\mathcal{F} \subseteq 2^{[n]}$ such that $\mathcal{F} \neq…

The Frankl's conjecture, formulated in 1979. and still open, states that in every family of sets closed for unions there is an element contained in at least half of the sets. A family Fc is called Frankl-complete (or FC-family) if in every…

A celebrated unresolved conjecture of Peter Frankl states that every finite collection of sets, with finite universe, admits an abundant element. In this paper, we prove Frankl's union-closed conjecture(FC). We provide an induction proof…

Given a finite poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced…

Given a finite poset $P$, we consider the largest size $\lanp$ of a family $\F$ of subsets of $[n]:=\{1,...,n\}$ that contains no subposet $P$. This continues the study of the asymptotic growth of $\lanp$; it has been conjectured that for…

We are interested in ordering the elements of a subset A of the non-zero integers modulo n in such a way that all the partial sums are distinct. We conjecture that this can always be done and we prove various partial results about this…

A family of sets is called union-closed if whenever $A$ and $B$ are sets of the family, so is $A\cup B$. The long-standing union-closed conjecture states that if a family of subsets of $[n]$ is union-closed, some element appears in at least…

F. Wehrung has asked: Given a family $\mathcal{C}$ of subsets of a set $\Omega$, under what conditions will there exist a total ordering on $\Omega$ under which every member of $\mathcal{C}$ is convex? <p> Note that if $A$ and $B$ are…

We prove that the conjecture made by Peter Frankl in the late 1970s is true. In other words for every finite union-closed family which contains a non?empty set, there is an element that belongs to at least half of its m

Assume $\mathcal{C}$ is the class of all linear orders $L$ such that $L$ is not a countable union of well ordered sets, and every uncountable subset of $L$ contains a copy of $\omega_1$. We show it is consistent that $\mathcal{C}$ has…

Let $P$ be a partially ordered set. We prove that if $n$ is sufficiently large, then there exists a packing $\mathcal{P}$ of copies of $P$ in the Boolean lattice $(2^{[n]},\subset)$ that covers almost every element of $2^{[n]}$:…

Let $(P,\leq)$ be a finite poset (partially ordered set), where $P$ has cardinality $n$. Consider linear extensions of $P$ as permutations $x_1x_2\cdots x_n$ in one-line notation. For distinct elements $x,y\in P$, we define…