2 research outputs found
Rainbow matchings in properly-colored hypergraphs
A hypergraph is properly colored if for every vertex , all the
edges incident to have distinct colors. In this paper, we show that if
, \cdots, are properly-colored -uniform hypergraphs on
vertices, where , and , then there exists a rainbow matching of size , containing one edge
from each . This generalizes some previous results on the Erd\H{o}s
Matching Conjecture
Simple juntas for shifted families
We say that a family of -element sets is a {\it -junta} if
there is a set of size such that, for any , its presence in
depends on its intersection with only. Approximating arbitrary
families by -juntas with small is a recent powerful technique in
extremal set theory.
The weak point of all known junta approximation results is that they work in
the range , where is an extremely fast growing function of the input
parameters, such as the quality of approximation or the number of families we
simultaneously approximate.
We say that a family is {\it shifted} if for any
and any such
that , we have . For many extremal set theory
problems, including the Erd\H os Matching Conjecture, or the Complete
-Intersection Theorem, it is sufficient to deal with shifted families only.
In this paper, we present very general approximation by juntas results for
shifted families with explicit (and essentially linear) dependency on the input
parameters. The results are best possible up to some constant factors.
Moreover, they give meaningful statements for almost all range of values of
. The proofs are shorter than the proofs of the previous approximation by
juntas results and are completely self-contained.
As an application of our junta approximation, we give a nearly-linear bound
for the multi-family version of the Erd\H os Matching Conjecture. More
precisely, we prove the following result. Let and suppose
that the families do
not contain such that 's
are pairwise disjoint. Then $\min_{i}|\mathcal F_i|\le {n\choose
k}-{n-s+1\choose k}.