35 research outputs found
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
Minimum Weight Flat Antichains of Subsets
Building on classical theorems of Sperner and Kruskal-Katona, we investigate
antichains in the Boolean lattice of all subsets of
, where is flat, meaning that it contains
sets of at most two consecutive sizes, say , where contains only -subsets,
while contains only -subsets. Moreover, we assume
consists of the first -subsets in squashed
(colexicographic) order, while consists of all -subsets
not contained in the subsets in . Given reals , we
say the weight of is
. We characterize the minimum
weight antichains for any given , and we do the
same when in addition is a maximal antichain. We can then derive
asymptotic results on both the minimum size and the minimum Lubell function
Towards a Geometric Approach to Strassen's Asymptotic Rank Conjecture
We make a first geometric study of three varieties in (for each ), including the Zariski
closure of the set of tight tensors, the tensors with continuous regular
symmetry. Our motivation is to develop a geometric framework for Strassen's
Asymptotic Rank Conjecture that the asymptotic rank of any tight tensor is
minimal. In particular, we determine the dimension of the set of tight tensors.
We prove that this dimension equals the dimension of the set of oblique
tensors, a less restrictive class introduced by Strassen.Comment: Final version. Revisions in Section 1 and Section
A characteristic free approach to secant varieties of triple Segre products
The goal of this short note is to study the secant varieties of the triple
Segre product of type (1,a,b) by means of the standard tools of combinatorial
commutative algebra. We reprove and extend to arbitrary characteristic results
of Landsberg and Weyman regarding the defining ideal and the Cohen-Macaulay
property of the secant varieties. Furthermore for these varieties we compute
the degree and give a bound for their Castelnuovo-Mumford regularity which is
sharp in many cases