A Note on Intersection Numbers of Difference Sets

We present a condition on the intersection numbers of difference sets which follows from a result of Jungnickel and Pott [3]. We apply this condition to rule out several putative (non-abelian) difference sets and to correct erroneous proofs of Lander [4] for the nonexistence of (352, 27, 2)- and (122, 37, 12)-difference sets

A Note on Intersection Numbers of Difference Sets

We present a condition on the intersection numbers of difference sets which follows from a result of Jungnickel and Pott [3]. We apply this condition to rule out several putative (non-abelian) difference sets and to correct erroneous proofs of Lander [4] for the nonexistence of (352, 27, 2)- and (122, 37, 12)-difference sets

Helly numbers of Algebraic Subsets of Rd\mathbb R^d

We study $S$-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in $\mathbb R^d$ with a proper subset $S\subset \mathbb R^d$. We contribute new results about their $S$-Helly numbers. We extend prior work for $S=\mathbb R^d$, $\mathbb Z^d$, and $\mathbb Z^{d-k}\times\mathbb R^k$; we give sharp bounds on the $S$-Helly numbers in several new cases. We considered the situation for low-dimensional $S$ and for sets $S$ that have some algebraic structure, in particular when $S$ is an arbitrary subgroup of $\mathbb R^d$ or when $S$ is the difference between a lattice and some of its sublattices. By abstracting the ingredients of Lov\'asz method we obtain colorful versions of many monochromatic Helly-type results, including several colorful versions of our own results.Comment: 13 pages, 3 figures. This paper is a revised version of what was originally the first half of arXiv:1504.00076v

Splittings of groups and intersection numbers

We prove algebraic analogues of the facts that a curve on a surface with self-intersection number zero is homotopic to a cover of a simple curve, and that two simple curves on a surface with intersection number zero can be isotoped to be disjoint.Comment: 40 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol4/paper6.abs.htm