5 research outputs found
Small doubling, atomic structure and -divisible set families
Let be a set family such that the intersection
of any two members of has size divisible by . The famous
Eventown theorem states that if then , and this bound can be achieved by, e.g., an `atomic'
construction, i.e. splitting the ground set into disjoint pairs and taking
their arbitrary unions. Similarly, splitting the ground set into disjoint sets
of size gives a family with pairwise intersections divisible by
and size . Yet, as was shown by Frankl and Odlyzko,
these families are far from maximal. For infinitely many , they
constructed families as above of size . On the other hand, if the intersection of {\em any number} of
sets in has size divisible by , then it is
easy to show that . In 1983 Frankl
and Odlyzko conjectured that holds
already if one only requires that for some any distinct members
of have an intersection of size divisible by . We
completely resolve this old conjecture in a strong form, showing that
if is chosen
appropriately, and the error term is not needed if (and only if) , and is sufficiently large. Moreover the only extremal
configurations have `atomic' structure as above. Our main tool, which might be
of independent interest, is a structure theorem for set systems with small
'doubling'
Additive Combinatorics
Abstract - A set family F that is a subset of 2^[n], [n]={1,...,n} is said to have the Eventown property if all its component sets are even sized and the intersection of any two of these sets is even sized. The Eventown theorem states a bound for the size of F in this case, namely |F| ≤ 2^[n/2]. The aim of the thesis is to discuss a generalization of the Eventown theorem through the lens of additive combinatorics
Exact k -Wise Intersection Theorems
A typical problem in extremal combinatorics is the following. Given a large number n and a set L, find the maximum cardinality of a family of subsets of a ground set of n elements such that the intersection of any two subsets has cardinality in L. We investigate the generalization of this problem, where intersections of more than 2 subsets are considered. In particular, we prove that when k−1 is a power of 2, the size of the extremal k-wise oddtown family is (k−1)(n− 2log2(k−1)). Tight bounds are also found in several other basic case
Two Remarks on Eventown and Oddtown Problems
ISSN:0895-4801ISSN:1095-714