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Graphs and Combinatorics 3, 91-94 (1987) Combinatorics

Summary: Graphs and Combinatorics 3, 91-94 (1987)
On the Kernel of Intersecting Families*
N. Alon 1.* and Z. Fiiredi 2
Department of Mathematics, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978 Israel
2 Mathematical Institute of the Hungarian Academy of Science, 1364 Budapest, P.O.B. 127,
Abstract.Let ~ be a t-wise s-intersecting family, i.e., [FaN"" NFtI > s holds for every t members of
~~.Then there exists a set Ysuch that IFx N.'- NF, N Y[ _>_s still holds for every Fl ..... F,e ~. Here
exponential lower and upper bounds are proven for the possible sizes of Y.
1. Intersecting Families
A family of sets ~ (or a hypergraph) is called intersecting if F N F' # ~ holds for
all F, F'~ ~-. The rank r(~) is defined by
r{~) =: max{IF]: F e ~,~}.
For a set Y define the restriction ~] Y of o~ to Y by
Y[Y =: {FN Y:Fe~}.
In 1964 Calczynska-Karlowicz [3] proved that for every k there exists an n(k) such
that for every intersecting hypergraph ~ of rank at most k there is a set Y of


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


Collections: Mathematics