A note on tilted Sperner families with patterns

Let $p$ and $q$ be two nonnegative integers with $p+q>0$ and $n>0$. We call $\mathcal{F} \subset \mathcal{P}([n])$ a \textit{(p,q)-tilted Sperner family with patterns on [n]} if there are no distinct $F,G \in \mathcal{F}$ with: $$(i) \ \ p|F \setminus G|=q|G \setminus F|, \ \textrm{and}$$ $$(ii) \ f > g \ \textrm{for all} \ f \in F \setminus G \ \textrm{and} \ g \in G \setminus F.$$ Long (\cite{L}) proved that the cardinality of a (1,2)-tilted Sperner family with patterns on $[n]$ is $$O(e^{120\sqrt{\log n}}\ \frac{2^n}{\sqrt{n}}).$$ We improve and generalize this result, and prove that the cardinality of every ($p,q$)-tilted Sperner family with patterns on [$n$] is $$O(\sqrt{\log n} \ \frac{2^n}{\sqrt{n}}).$$Comment: 8 page

Generalized Tur\'an problems for disjoint copies of graphs

Given two graphs $H$ and $F$, the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices is denoted by $ex(n,H,F)$. We investigate the function $ex(n,H,kF)$, where $kF$ denotes $k$ vertex disjoint copies of a fixed graph $F$. Our results include cases when $F$ is a complete graph, cycle or a complete bipartite graph.Comment: 18 pages. There was a wrong statement in the first version, it is corrected no

A discrete isodiametric result: the Erd\H{o}s-Ko-Rado theorem for multisets

There are many generalizations of the Erd\H{o}s-Ko-Rado theorem. We give new results (and problems) concerning families of $t$-intersecting $k$-element multisets of an $n$-set and point out connections to coding theory and classical geometry. We establish the conjecture that for $n \geq t(k-t)+2$ such a family can have at most ${n+k-t-1\choose k-t}$ members

Forbidden subposet problems for traces of set families

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 $i:P\rightarrow \{F_1,F_2, \dots,F_{|P|}\}$ such that for any $p,p'\in P$ the relation $p<_P p'$ implies $i(p)\subsetneq i(p')$. A family $\mathcal{F}$ of sets is \textit{$P$-free} if it does not contain any copy of $P$. The trace of a family $\mathcal{F}$ on a set $X$ is $\mathcal{F}|_X:=\{F\cap X: F\in \mathcal{F}\}$. We introduce the following notions: $\mathcal{F}\subseteq 2^{[n]}$ is $l$-trace $P$-free if for any $l$-subset $L\subseteq [n]$, the family $\mathcal{F}|_L$ is $P$-free and $\mathcal{F}$ is trace $P$-free if it is $l$-trace $P$-free for all $l\le n$. As the first instances of these problems we determine the maximum size of trace $B$-free families, where $B$ is the butterfly poset on four elements $a,b,c,d$ with $a,b<c,d$ and determine the asymptotics of the maximum size of $(n-i)$-trace $K_{r,s}$-free families for $i=1,2$. We also propose a generalization of the main conjecture of the area of forbidden subposet problems

Rounds in a combinatorial search problem

We consider the following combinatorial search problem: we are given some excellent elements of $[n]$ and we should find at least one, asking questions of the following type: "Is there an excellent element in $A \subset [n]$?". G.O.H. Katona proved sharp results for the number of questions needed to ask in the adaptive, non-adaptive and two-round versions of this problem. We verify a conjecture of Katona by proving that in the $r$-round version we need to ask $rn^{1/r}+O(1)$ queries for fixed $r$ and this is sharp. We also prove bounds for the queries needed to ask if we want to find at least $d$ excellent elements.Comment: 14 page