 
Summary: Graphs and Combinatorics 3, 9194 (1987)
Graphsand
Combinatorics
© SpringerVerlag1987
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,
Hungary
Abstract.Let ~ be a twise sintersecting 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 CalczynskaKarlowicz [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
