Paper Info
Title
Families that Remain k-Sperner Even After Omitting an Element of their Ground Set.
Abstract
A family F subset of 2(inverted right perpendicular n inverted right perpendicular) of sets is said to be l-trace k-Serner if for any l-subset L subset of [n] the family F vertical bar(L) = {F vertical bar(L) : F is an element of F} = {F boolean AND L : F is an element of F} is k-Sperner, i.e does not contain any chain of length k+1. The maximum size that an l-trace k-Sperner family F subset of 2([n]) can have is denoted by f(n, k, l). For pairs of integers l < k, if in a family G every pair of sets satisfies parallel to G(1)vertical bar - vertical bar G(2)parallel to < k-1, then G possesses the (n - l)-trace k-Sperner property. Among such families, the large stone is F-0 - {F is an element of 2([n]) : left perpendicular n-(k-l)/2 right perpendicular + 1 <= vertical bar F vertical bar <= left perpendicular n-(k-l)/2 right perpendicular + k - l} and also F'(0) = {F is an element of 2([n]) : left perpendicular n-(k-l)/2 right perpendicular <= vertical bar F vertical bar <= left perpendicular n-(k-l)/2 right perpendicular + k - l - 1} if n - (k - 1) is even. In an earlier paper, we proved that this is asymptotically optimal for all pair of integer l < k, i.e f(n, k, n - 1) = (1 + o(1)vertical bar F-0 vertical bar. In this paper we consider the case when l = 1, k >= 2,and prove that f(n, k, n - 1)=vertical bar F-0 vertical bar provided n is large enough. We also prove that the unique (n - 1)-trace k-Sperner family with size f(n, k, n - 1)is F-0 and also F-0' when n + k is odd.
Year
Venue
DocType
2013
ELECTRONIC JOURNAL OF COMBINATORICS
Journal
Volume
Issue
ISSN
20.0
1.0
1077-8926
Citations 
PageRank 
References 
0
0.34
0
Authors
1
Name
Order
Citations
PageRank
Balázs Patkós18521.60