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Economical covers with geometric applications Bela Bollobas
 

Summary: Economical covers with geometric applications
Noga Alon
B´ela Bollob´as
Jeong Han Kim
Van H. Vu §
Abstract
Given a hypergraph H, a cover of H is a collection of edges whose union is the set of vertices;
the minimal number of edges in a cover is the covering number cov(H) of H. The maximal codegree
2(H) is the maximal number of edges containing two fixed vertices of H. For D = 1, 2, . . .,
let HD be a D-regular k-uniform hypergraph on n vertices, where k and n are functions of D.
Among other results, we shall prove that if 2(HD) = o(D/e2k
log D) and k = o(log D) then
cov(HD) = (1 + o(1))n/k; this extends the known result that this holds for fixed k. On the other
hand, if k 4 log D then cov(HD) (n
k log( k
log D )) may hold even when 2(HD) = 1. Several
extensions and variants are also obtained, as well as the following geometric application. The
minimum number of lines required to separate n random points in the unit square is, almost surely,
(n2/3
/(log n)1/3

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics