Chapter 1

Our proof of the Main Theorem starts by assuming, on the contrary, that there exists a

neighborly family F in

E3,

consisting of nine tetrahedra P1? P2, ... , P9. Let {H1? ..., Ht}

denote the collection of all the planes in E3 which contain facets of members of F. Let H+j and

H"j denote the two closed half-spaces, determined by Hj, for all j .

Following Baston [2] (see also [17, 12, 20, 21, 23]), let B(F) = (by) be the Baston matrix

of F, defined by

/ +1 if H. contains a facet of P and P c

H+

,

I J i 1 j

b = - 1 if H. contains a facet of P and P. c H7 ,

ij I J i 1 J

V 0 otherwise,

for all l i 9 and 1 jt.

The following properties hold:

(1) Each row of B(F) has precisely four nonzero terms.

(2) For every pair of row-indices i and j , there exists a unique column-index k, such that

bik-bjk = - 1 -

Property (1) follows from the fact that every tetrahedron has four facets, while property (2)

is implied by the neighborliness of F, see [2,17].

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