9,814 research outputs found
On the number of maximal intersecting k-uniform families and further applications of Tuza's set pair method
We study the function which denotes the number of maximal
-uniform intersecting families . Improving a
bound of Balogh at al. on , we determine the order of magnitude of
by proving that for any fixed , holds. Our proof is based on Tuza's set pair
approach.
The main idea is to bound the size of the largest possible point set of a
cross-intersecting system. We also introduce and investigate some related
functions and parameters.Comment: 11 page
Vertex covers by monochromatic pieces - A survey of results and problems
This survey is devoted to problems and results concerning covering the
vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles
and other objects. It is an expanded version of the talk with the same title at
the Seventh Cracow Conference on Graph Theory, held in Rytro in September
14-19, 2014.Comment: Discrete Mathematics, 201
Nonnegative k-sums, fractional covers, and probability of small deviations
More than twenty years ago, Manickam, Mikl\'{o}s, and Singhi conjectured that
for any integers satisfying , every set of real numbers
with nonnegative sum has at least -element subsets whose
sum is also nonnegative. In this paper we discuss the connection of this
problem with matchings and fractional covers of hypergraphs, and with the
question of estimating the probability that the sum of nonnegative independent
random variables exceeds its expectation by a given amount. Using these
connections together with some probabilistic techniques, we verify the
conjecture for . This substantially improves the best previously
known exponential lower bound . In addition we prove
a tight stability result showing that for every and all sufficiently large
, every set of reals with a nonnegative sum that does not contain a
member whose sum with any other members is nonnegative, contains at least
subsets of cardinality with
nonnegative sum.Comment: 15 pages, a section of Hilton-Milner type result adde
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