610 research outputs found
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
Minimizing the regularity of maximal regular antichains of 2- and 3-sets
Let be a natural number. We study the problem to find the
smallest such that there is a family of 2-subsets and
3-subsets of with the following properties: (1)
is an antichain, i.e. no member of is a subset of
any other member of , (2) is maximal, i.e. for every
there is an with or , and (3) is -regular, i.e. every point
is contained in exactly members of . We prove lower
bounds on , and we describe constructions for regular maximal antichains
with small regularity.Comment: 7 pages, updated reference
Maximal antichains of minimum size
Let be a natural number, and let be a set . We study the problem to find the smallest possible size of a
maximal family of subsets of such that
contains only sets whose size is in , and for all
, i.e. is an antichain. We present a
general construction of such antichains for sets containing 2, but not 1.
If our construction asymptotically yields the smallest possible size
of such a family, up to an error. We conjecture our construction to be
asymptotically optimal also for , and we prove a weaker bound for
the case . Our asymptotic results are straightforward applications of
the graph removal lemma to an equivalent reformulation of the problem in
extremal graph theory which is interesting in its own right.Comment: fixed faulty argument in Section 2, added reference
On the Duality of Semiantichains and Unichain Coverings
We study a min-max relation conjectured by Saks and West: For any two posets
and the size of a maximum semiantichain and the size of a minimum
unichain covering in the product are equal. For positive we state
conditions on and that imply the min-max relation. Based on these
conditions we identify some new families of posets where the conjecture holds
and get easy proofs for several instances where the conjecture had been
verified before. However, we also have examples showing that in general the
min-max relation is false, i.e., we disprove the Saks-West conjecture.Comment: 10 pages, 3 figure
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