A conjecture of V. Sós [3] is proved that any set of 3 4 n 3 + cn 2 triples from an n-set, where c is a suitable absolute constant, must contain a copy of the Fano configuration (the projective plane of order two). This is an asymptotically sharp estimate. Given a 3-uniform hypergraph F, let ex 3 (n, F) denote the maximum possible size of a 3-uniform hypergraph of order n that does not contain any subhypergraph isomorphic to F. Our terminology follows that of [1], which is a comprehensive survey of Turántype extremal problems. An elementary and well known averaging argument shows that the ratio ex 3 (n, F)/ n 3 is a non-increasing sequence, so that π(F) := lim n→∞ ex 3 (n, F)/ n 3 exists. Theorem. If F = P G(2, 2) is the Fano plane then π(F) = 3 4 .

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A construction is given of ${{2}\over {9}} (d+1)^2$ equiangular lines in Euclidean $d$-space, when $d = 3 \cdot 2^{2t-1}-1$ with $t$ any positive integer. This compares with the well known "absolute" upper bound of ${{1}\over {2}} d(d+1)$ lines in any equiangular set; it is the first known constructive lower bound of order $d^2$ .

A construction of association schemes is presented; these are fission schemes of the triangular schemes T (n) where n = q + 1 with q any prime power. The key observation is quite elementary, being that the natural action of P G L(2, q) on the 2-element subsets of the projective line P G(1, q) is generously transitive. In addition, some observations on the intersection parameters and fusion schemes of these association schemes are made. THE CONSTRUCTIONThis paper is a sequel to [4]. In that paper, it was observed that almost all known selfdual classical association schemes have natural fission schemes (fissioning the maximumdistance relation only); whereas in the non-self-dual case there seemed to be no analogous fission schemes. Subsequently, we found that there is at least one such non-self-dual classical association scheme that admits an interesting fission scheme, namely the triangular scheme T (n) = J (n, 2) where n = q + 1 with q any prime power; this is the object of the present work. For terminology and background, we refer to Bannai and Ito [2] for association schemes and to Hirschfeld [9] for finite geometry. Recall that the group P G L(2, q) acts (as Möbius transformations) on the projective line P G(1, q); this action is (sharply) 3-transitive. There is a natural induced action on the 2-element subsets of the projective line, (2, q). In the proof below we apply the basic fact (cf. [9, p. 135]) that the cross-ratiois a complete invariant for ordered quadruples of distinct points on the projective line, i.e., one quadruple may be mapped to another quadruple (via a Möbius transformation) if and only if they have the same cross-ratio. THEOREM. The action of P G L(2, q) on the 2-element subsets of P G(1, q) is generously transitive.PROOF. Given intersecting 2-sets {a, b} and {a, c}, there is some M in P G L(2, q) that swaps them, since the group is triply transitive. And given disjoint 2-sets {a, b} and {c, d}, there is also some Möbius transformation that interchanges them, because the ordered quadruples (a, b, c, d) and (c, d, a, b) have the same cross-ratio.2Given any transitive permutation group G acting on a set , the orbitals are the orbits in × under the natural action of G on pairs. If G is generously transitive, then the orbitals form the relations (associate classes) of a symmetric association scheme (cf. [2, p. 54]). In our case, the relations can be described as follows. One relation, say R 1 , is the line-graph of the complete graph (i.e., one relation of the triangular scheme T (q + 1) has remained unfissioned). Next, for each reciprocal pair {s, s −1 } of elements in G F(q)\{0, 1}, there is a relation R {s,s −1 } where {a, b} and {c, d} are in this relation when ρ (a, b, c, d) equals s or s −1 . Note that ρ (b, a, c, d) = ρ(a, b, c, d) −1 so this makes sense as a definition for unordered