TY - JOUR

T1 - A construction of almost Steiner systems

AU - Ferber, Asaf

AU - Hod, Rani

AU - Krivelevich, Michael

AU - Sudakov, Benny

N1 - Publisher Copyright:
© 2013 Wiley Periodicals, Inc.

PY - 2014/11

Y1 - 2014/11

N2 - Let n, k, and t be integers satisfying n > k > t ≥ 2. A Steiner system with parameters t, k, and n is a k-uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for t ≥ 6. In this note we prove that for every k > t ≥ 2 and sufficiently large n, there exists an almost Steiner system with parameters t, k, and n; that is, there exists a k-uniform hypergraph on n vertices such that every set of t distinct vertices is covered by either one or two edges.

AB - Let n, k, and t be integers satisfying n > k > t ≥ 2. A Steiner system with parameters t, k, and n is a k-uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for t ≥ 6. In this note we prove that for every k > t ≥ 2 and sufficiently large n, there exists an almost Steiner system with parameters t, k, and n; that is, there exists a k-uniform hypergraph on n vertices such that every set of t distinct vertices is covered by either one or two edges.

KW - Steiner systems

UR - http://www.scopus.com/inward/record.url?scp=84908231065&partnerID=8YFLogxK

U2 - 10.1002/jcd.21380

DO - 10.1002/jcd.21380

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AN - SCOPUS:84908231065

SN - 1063-8539

VL - 22

SP - 488

EP - 494

JO - Journal of Combinatorial Designs

JF - Journal of Combinatorial Designs

IS - 11

ER -