By J. Srivastava (Eds.)

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We will completely determine the A-quadrangles with A > 2. 1. A partial A-geometry will be called extremal if the condition lGnHHnKI>1 for blocks G, H, K implies that (GHnHflKIaA. 2. The inequality e 2 A* - A + 1 holds in every extremal partial A-geometry 3. Equality holds if and only i f the following criterion is satisfied: for each nonf7ag ( p , G), the e blocks through p which intersect G induce a projective plane of order A - 1 on the e points of G which are joined to p (degenerate unless A >2).

Deza 52 all its elements of height n is a sharp A(L, n ) and (I, -L l ) K L I - L 2 ) I . . K k 4)l 1,. 4. Any sharp A = A(>n - t, n ) is a set of all elements of height n of a PG(L,n), Q with L = { n - t + l , . . , n } . In fact we can take Q = {c E 9’:either c E A or 121s t - 1). e. trivoid in terminology of PMD’s). So we have to prove the condition (c) in the definition of PG(L, n ) only for a with [lull= t - 1 = n - 1, = 1: But this condition for our case just consists of saying that for any subpermutation a of height t there exists exactly one C E Awith a < c .

Partial A-geometries of small nexus 23 then kj[2A2- ( j + l ) ( h+ e - 1 ) ] GA 4 - 3h3+ 2A2+4jh2- h ( e + l)(j”+j). (4) Then if 2A2- (j + l ) ( h+ e - 1 ) > 0 , we may replace k by ( e + 1) in (4), obtaining a strict inequality. Restating, we have ( e + 1)[2jh2- (j’ + j ) ( A + e - l ) ]j . h+e-1 Now (5) simplifies to O<(h4- 3h3+ 2h2)+j(-2eh2+ 2h2+e 2 - 1 ) + j 2 ( e 2 -1). (7) Consider the quantity d + 1). =h2/(e (8) We wish to substitute [ d ] , the largest integer

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