16 research outputs found
On Hamiltonian cycles in hypergraphs with dense link graphs
We show that every -uniform hypergraph on vertices whose minimum
-degree is at least contains a Hamiltonian cycle. A
construction due to Han and Zhao shows that this minimum degree condition is
optimal. The same result was proved independently by Lang and Sahueza-Matamala.Comment: Dedicated to Endre Szemer\'edi on the occasion of his 80th birthda
-cross -intersecting families via necessary intersection points
Given integers and we call families
-cross
-intersecting if for all , , we have
. We obtain a strong generalisation of
the classic Hilton-Milner theorem on cross intersecting families. In
particular, we determine the maximum of for -cross -intersecting families in the
cases when these are -uniform families or arbitrary subfamilies of
. Only some special cases of these results had been proved
before. We obtain the aforementioned theorems as instances of a more general
result that considers measures of -cross -intersecting families. This
also provides the maximum of for
families of possibly mixed uniformities .Comment: 13 page
On extremal problems concerning the traces of sets
Given two non-negative integers and , define to be the
maximal number such that in every hypergraph on vertices and
with at most edges there is a vertex such that
, where
. This problem has been
posed by F\"uredi and Pach and by Frankl and Tokushige. While the first results
were only for specific small values of , Frankl determined
for all with . Subsequently, the goal became to
determine for larger . Frankl and Watanabe determined
for . Other general results were not known so
far.
Our main result sheds light on what happens further away from powers of two:
We prove that for and
and give an example showing that this equality does not hold for . The
other line of research on this problem is to determine for small
values of . In this line, our second result determines for
. This solves more instances of the problem for small and in
particular solves a conjecture by Frankl and Watanabe.Comment: 16 page