Twelve additional chapters discuss equivariant minimal Langrange surfaces, Multibrot sets, the Matovich-Pearson equations, diffeomorphisms with special properties, Euler products, application of the Epstein zeta function to hexagonal lattices in physics, an attempt to find q-analogues of the

Fano Plane resulting in a record-sized subspace code, integral orthogonal groups, Fourier analysis in X-ray crystallography, a new proof for joint continuity of semiflows, a convergent string method for approximating the Hamiltonian, and pluri-Lagrangian systems.

In fact, the link at every vertex in such a wall is a spherical building of type [A.sub.2], namely, the incidence graph of the Fano plane. The spectral properties of this graph make it a "good" expander--it is a Ramanujan graph--and yet, it will have to be cut into two large subsets without discarding too many edges during the proof.

In addition, the link L of z is the incidence graph of the Fano plane. The path [[gamma].sub.t], for t > [t.sub.0] small enough, provides two points [a.sub.t] and [b.sub.t] in L, which are connected in L \ W for t > [t.sub.0] but are not connected in L \ W for t = [t.sub.0].

In [17] we studied Forb(n, F), where F is the 3-uniform hypergraph of the Fano plane, which is the unique triple system with 7 hyperedges on 7 vertices where every pair of vertices is contained in precisely one hyperedge.

Since the Fano plane is linear, this counting lemma is applicable.

where by F we will always denote the hypergraph of the Fano plane. We will refer to hypergraphs not containing a copy of F, as Fano-free hypergraphs.

We recall the result of Furedi and Simonovits [8] and Keevash and Sudakov [11] on the extremal number for the hypergraph of the Fano plane F asserts for sufficiently large n:

The following stability result for Fano-free hypergraphs was proved by Keevash and Sudakov [11] and Furedi and Simonovits [8], in order to determine the extremal hypergraph for the Fano plane:

Otherwise, any hyperedge e, say in [X.sub.H], together with the [K.sub.4] in [Y.sub.H], which lies in the common link of the vertices of e would span a copy of the hypergraph of the Fano plane.

Later we will apply the Key-lemma, Theorem 5, with l = 7 and L being the hypergraph of the Fano plane to the cluster-hypergraph of an [epsilon]-regular partition.

It must be Fano-free as otherwise Theorem 5 would imply that H also contains a copy of the hypergraph of the Fano plane.

More precisely, we will assume that there exists H [member of] [F''.sub.n] ([alpha], [beta]) with max{[DELTA](H[[X.sub.H]]), [DELTA](H[[Y.sub.H]])} [greater than or equal to] [gamma][n.sup.2], and we will show that H contains a Fano plane.