18 research outputs found
Upper bounds for sunflower-free sets
A collection of sets is said to form a -sunflower, or -system,
if the intersection of any two sets from the collection is the same, and we
call a family of sets sunflower-free if it contains no
sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt and
Croot, Lev and Pach we apply the polynomial method directly to
Erd\H{o}s-Szemer\'{e}di sunflower problem and prove that any sunflower-free
family of subsets of has size at most We say that
a set for is
sunflower-free if every distinct triple there exists a coordinate
where exactly two of are equal. Using a version of the
polynomial method with characters
instead of polynomials, we
show that any sunflower-free set has size
where . This can be
seen as making further progress on a possible approach to proving the
Erd\H{o}s-Rado sunflower conjecture, which by the work of Alon, Sphilka and
Umans is equivalent to proving that for some constant
independent of .Comment: 5 page
2-cancellative hypergraphs and codes
A family of sets F (and the corresponding family of 0-1 vectors) is called
t-cancellative if for all distict t+2 members A_1,... A_t and B,C from F the
union of A_1,..., A_t and B differs from the union of A_1, ..., A_t and C. Let
c(n,t) be the size of the largest t-cancellative family on n elements, and let
c_k(n,t) denote the largest k-uniform family. We significantly improve the
previous upper bounds, e.g., we show c(n,2) n_0). Using an
algebraic construction we show that the order of magnitude of c_{2k}(n,2) is
n^k for each k (when n goes to infinity).Comment: 20 page
Upper Bounds For Families Without Weak Delta-Systems
For , a collection of sets is said to form a \emph{weak
-system} if the intersection of any two sets from the collection has
the same size. Erd\H{o}s and Szemer\'{e}di asked about the size of the largest
family of subsets of that does not contain a weak
-system. In this note we improve upon the best upper bound of the
author and Sawin from arXiv:1606.09575 and show that where
is the capset capacity. In particular, this shows that Comment: 6 pages. Minor change
Improved bounds for the sunflower lemma
A sunflower with petals is a collection of sets so that the
intersection of each pair is equal to the intersection of all. Erd\H{o}s and
Rado proved the sunflower lemma: for any fixed , any family of sets of size
, with at least about sets, must contain a sunflower. The famous
sunflower conjecture is that the bound on the number of sets can be improved to
for some constant . In this paper, we improve the bound to about
. In fact, we prove the result for a robust notion of sunflowers,
for which the bound we obtain is tight up to lower order terms.Comment: Revised preprint, added sections on applications and rainbow
sunflower
Odd-Sunflowers
Extending the notion of sunflowers, we call a family of at least two sets an
odd-sunflower if every element of the underlying set is contained in an odd
number of sets or in none of them. It follows from the Erd\H os--Szemer\'edi
conjecture, recently proved by Naslund and Sawin, that there is a constant
such that every family of subsets of an -element set that contains
no odd-sunflower consists of at most sets. We construct such families
of size at least . We also characterize minimal odd-sunflowers of
triples
Sunflowers in Set Systems of Bounded Dimension
Given a family of -element sets, form an {\em -sunflower} if for all and . According to a famous conjecture of Erd\H os and Rado
(1960), there is a constant such that if , then
contains an -sunflower.
We come close to proving this conjecture for families of bounded {\em
Vapnik-Chervonenkis dimension}, VC-dim. In this case, we
show that -sunflowers exist under the slightly stronger assumption
. Here, denotes the iterated
logarithm function.
We also verify the Erd\H os-Rado conjecture for families of
bounded {\em Littlestone dimension} and for some geometrically defined set
systems