Translation planes of order 23^2

We give a complete classication of translation planes of order 23^2 whose translation complement contains a subgroup G such that the quotient group G modulo scalars is isomorphic to A_6. Up to isomorphisms, there are exactly 23 such planes and six of them have a larger translation complement being modulo scalars isomorphic to S_6

Intriguing sets of strongly regular graphs and their related structures

In this paper we outline a technique for constructing directed strongly regular graphs by using strongly regular graphs having a "nice" family of intriguing sets. Further, we investigate such a construction method for rank three strongly regular graphs having at most $45$ vertices. Finally, several examples of intriguing sets of polar spaces are provided

Tight sets in finite classical polar spaces

We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries PG(2r + 1, root q) and generators of H(2r + 1, q), if q >= 81 is an odd square and i < (q(2/3) - 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Delta,Delta(perpendicular to)}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Delta,Delta(perpendicular to)} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity perpendicular to of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under perpendicular to. This improves previous results where i < q(5/8)/root 2+ 1 was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q(2/3) - 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q)

On 4-general sets in finite projective spaces

A $4$-general set in ${\rm PG}(n,q)$ is a set of points of ${\rm PG}(n,q)$ spanning the whole ${\rm PG}(n,q)$ and such that no four of them are on a plane. Such a pointset is said to be complete if it is not contained in a larger $4$-general set of ${\rm PG}(n, q)$. In this paper upper and lower bounds for the size of the largest and the smallest complete $4$-general set in ${\rm PG}(n,q)$, respectively, are investigated. Complete $4$-general sets in ${\rm PG}(n,q)$, $q \in \{3,4\}$, whose size is close to the theoretical upper bound are provided. Further results are also presented, including a description of the complete $4$-general sets in projective spaces of small dimension over small fields and the construction of a transitive $4$-general set of size $3(q + 1)$ in ${\rm PG}(5, q)$, $q \equiv 1 \pmod{3}$

Singer quadrangles

[no abstract available

Characterisations of finite egglike inversive planes

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