7 research outputs found
Decomposing 1-Sperner hypergraphs
A hypergraph is Sperner if no hyperedge contains another one. A Sperner
hypergraph is equilizable (resp., threshold) if the characteristic vectors of
its hyperedges are the (minimal) binary solutions to a linear equation (resp.,
inequality) with positive coefficients. These combinatorial notions have many
applications and are motivated by the theory of Boolean functions and integer
programming. We introduce in this paper the class of -Sperner hypergraphs,
defined by the property that for every two hyperedges the smallest of their two
set differences is of size one. We characterize this class of Sperner
hypergraphs by a decomposition theorem and derive several consequences from it.
In particular, we obtain bounds on the size of -Sperner hypergraphs and
their transversal hypergraphs, show that the characteristic vectors of the
hyperedges are linearly independent over the reals, and prove that -Sperner
hypergraphs are both threshold and equilizable. The study of -Sperner
hypergraphs is motivated also by their applications in graph theory, which we
present in a companion paper
On -close Sperner systems
For a set of positive integers, a set system is said to be -close Sperner, if for any pair of distinct
sets in the skew distance belongs to . We reprove an extremal result of Boros,
Gurvich, and Milani\v c on the maximum size of -close Sperner set systems
for and generalize to and obtain slightly weaker bounds for
arbitrary . We also consider the problem when might include 0 and
reprove a theorem of Frankl, F\"uredi, and Pach on the size of largest set
systems with all skew distances belonging to
Sperner systems with restricted differences
Let be a family of subsets of and be a subset of
. We say is an -differencing Sperner system if
for any distinct . Let be a prime
and be a power of . Frankl first studied -modular -differencing
Sperner systems and showed an upper bound of the form
. In this paper, we obtain new upper bounds on
-modular -differencing Sperner systems using elementary -adic analysis
and polynomial method, extending and improving existing results substantially.
Moreover, our techniques can be used to derive new upper bounds on subsets of
the hypercube with restricted Hamming distances. One highlight of the paper is
the first analogue of the celebrated Snevily's theorem in the -modular
setting, which results in several new upper bounds on -modular -avoiding
-intersecting systems. In particular, we improve a result of Felszeghy,
Heged\H{u}s, and R\'{o}nyai, and give a partial answer to a question posed by
Babai, Frankl, Kutin, and \v{S}tefankovi\v{c}.Comment: 22 pages, results in table 1 and section 6.1 improve