Abstract
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 $P$-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.
Highlights
In this paper we introduce a problem that bridges two areas of extremal finite set theory, namely forbidden subposet problems and traces of set families
There is a vast literature of Sperner type problems, we focus on forbidden subposet problems introduced by Katona and Tarjan
A family F of sets is P -free if it does not contain any copy of P
Summary
In this paper we introduce a problem that bridges two areas of extremal finite set theory, namely forbidden subposet problems and traces of set families. For any poset P let e(P ) denote the largest integer k such that for any j and n the family. The second author proved in [13] that the only k-trace Pk+1-free families are (he showed that for any fixed k ≤ l we have T rl(n, Pk+1) =. Note that for any poset P if y(P ) denotes the largest integer m with 2[m] not containing a copy of P , by Theorem 1.3 we have T r(n, P ) ≤. The following theorem implies Conjecture 1.6 in the case of the posets Kr,s. The remainder of this paper is organized as follows: Section 2 deals with trace P -free families, Theorem 1.5 along with some further remarks are shown there.
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