243 research outputs found
Regular Intersecting Families
We call a family of sets intersecting, if any two sets in the family
intersect. In this paper we investigate intersecting families of
-element subsets of such that every element of
lies in the same (or approximately the same) number of members of
. In particular, we show that we can guarantee if and only if .Comment: 15 pages, accepted versio
The minimum number of nonnegative edges in hypergraphs
An r-unform n-vertex hypergraph H is said to have the
Manickam-Mikl\'os-Singhi (MMS) property if for every assignment of weights to
its vertices with nonnegative sum, the number of edges whose total weight is
nonnegative is at least the minimum degree of H. In this paper we show that for
n>10r^3, every r-uniform n-vertex hypergraph with equal codegrees has the MMS
property, and the bound on n is essentially tight up to a constant factor. This
result has two immediate corollaries. First it shows that every set of n>10k^3
real numbers with nonnegative sum has at least nonnegative
k-sums, verifying the Manickam-Mikl\'os-Singhi conjecture for this range. More
importantly, it implies the vector space Manickam-Mikl\'os-Singhi conjecture
which states that for n >= 4k and any weighting on the 1-dimensional subspaces
of F_q^n with nonnegative sum, the number of nonnegative k-dimensional
subspaces is at least . We also discuss two additional
generalizations, which can be regarded as analogues of the Erd\H{o}s-Ko-Rado
theorem on k-intersecting families
Covering the complete graph by partitions
AbstractA (D, c)-coloring of the complete graph Kn is a coloring of the edges with c colors such that all monochromatic connected subgraphs have at most D vertices. Resolvable block designs with c parallel classes and with block size D are natural examples of (D, c)-colorings. However, (D, c)-colorings are more relaxed structures. We investigate the largest n such that Kn has a (D, c)-coloring. Our main tool is the fractional matching theory of hypergraphs
Covering pairs by q2 + q + 1 sets
AbstractFor given k and s let n(k, s) be the largest cardinality of a set whose pairs can be covered by sk-sets. We determine n(k, q2 + q + 1) if a PG(2, q) exists, k > q(q + 1)2, and the remainder of k divided by (q + 1) is at least √q. Asymptotic results are also given for n(k, s) whenever s is fixed and k → ∞. Our main tool is the theory of fractional matchings of hypergraphs
Combinatorial problems in finite geometry and lacunary polynomials
We describe some combinatorial problems in finite projective planes and
indicate how R\'edei's theory of lacunary polynomials can be applied to them
A Universal Homogeneous Simple Matroid of Rank
We construct a -homogeneous universal simple matroid of rank ,
i.e. a countable simple rank~ matroid which -embeds every
finite simple rank matroid, and such that every isomorphism between finite
-subgeometries of extends to an automorphism of . We also
construct a -homogeneous matroid which is universal for the
class of finite simple rank matroids omitting a given finite projective
plane . We then prove that these structures are not -categorical,
they have the independence property, they admit a stationary independence
relation, and that their automorphism group embeds the symmetric group
. Finally, we use the free projective extension of
to conclude the existence of a countable projective plane embedding all the
finite simple matroids of rank and whose automorphism group contains
, in fact we show that
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