128,218 research outputs found
Comparable pairs in families of sets
Given a family of subsets of , we say two sets are comparable if or . Sperner's
celebrated theorem gives the size of the largest family without any comparable
pairs. This result was later generalised by Kleitman, who gave the minimum
number of comparable pairs appearing in families of a given size.
In this paper we study a complementary problem posed by Erd\H{o}s and Daykin
and Frankl in the early '80s. They asked for the maximum number of comparable
pairs that can appear in a family of subsets of , a quantity we denote
by . We first resolve an old conjecture of Alon and Frankl, showing
that when . We also obtain more
accurate bounds for for sparse and dense families, characterise the
extremal constructions for certain values of , and sharpen some other known
results.Comment: 18 page
Colouring set families without monochromatic k-chains
A coloured version of classic extremal problems dates back to Erd\H{o}s and
Rothschild, who in 1974 asked which -vertex graph has the maximum number of
2-edge-colourings without monochromatic triangles. They conjectured that the
answer is simply given by the largest triangle-free graph. Since then, this new
class of coloured extremal problems has been extensively studied by various
researchers. In this paper we pursue the Erd\H{o}s--Rothschild versions of
Sperner's Theorem, the classic result in extremal set theory on the size of the
largest antichain in the Boolean lattice, and Erd\H{o}s' extension to
-chain-free families.
Given a family of subsets of , we define an
-colouring of to be an -colouring of the sets without
any monochromatic -chains . We
prove that for sufficiently large in terms of , the largest
-chain-free families also maximise the number of -colourings. We also
show that the middle level, , maximises the
number of -colourings, and give asymptotic results on the maximum
possible number of -colourings whenever is divisible by three.Comment: 30 pages, final versio
On pairs of definable orthogonal families
We introduce the notion of an M-family of infinite subsets of \nn which is
implicitly contained in the work of A. R. D. Mathias. We study the structure of
a pair of orthogonal hereditary families \aaa and \bbb, where \aaa is
analytic and \bbb is -measurable and an M-family.Comment: 21 pages, no figures. Illinois Journal of Mathematics (to appear
A Special Class of Almost Disjoint Families
The collection of branches (maximal linearly ordered sets of nodes) of the
tree (ordered by inclusion) forms an almost disjoint
family (of sets of nodes). This family is not maximal -- for example, any level
of the tree is almost disjoint from all of the branches. How many sets must be
added to the family of branches to make it maximal? This question leads to a
series of definitions and results: a set of nodes is {\it off-branch} if it is
almost disjoint from every branch in the tree; an {\it off-branch family} is an
almost disjoint family of off-branch sets; {\frak o}=\min\{|{\Cal O}|: {\Cal
O} is a maximal off-branch family. Results concerning include:
(in ZFC) , and (consistent with ZFC) is not
equal to any of the standard small cardinal invariants , ,
, or . Most of these consistency results use
standard forcing notions -- for example, comes from starting with a model of and
adding -many Cohen reals. Many interesting open questions remain,
though -- for example,
- …