# Random stuff

Unpublished fragments of work inspired by Ph.D. / M.Sc. theses of people nearby:
Man, Erik, Tim, Ralf, Sander, Jochem ...

## Man

Man V. M. Nguyen wrote a thesis
("Computer-algebraic methods for the construction of designs of experiments") among other
things about orthogonal arrays of strength 3. Part of that work was published:
A.E. Brouwer, A.M. Cohen, M.V.M. Nguyen,
*Fractional factorial designs of strength 3 and small run sizes*,
J. Statist. Planning and Inference 136 (2006) 3268-3280.
preprint

Some data:
Small orthogonal arrays of strength 3.

## Erik

Erik Postma wrote a
thesis
("Lie Algebras to Geometry and Back").
The note below proves a conjecture from Chapter 4 of that thesis.
Uniqueness of Lie orientation of Lie-orientable partial linear spaces.

(This note was incorporated in a 5-author draft

A.E. Brouwer, A.M. Cohen, H. Cuypers, J.I. Hall & E. Postma,
*Lie algebras, 2-groups and cotriangular spaces*,
Preprint
July 2009.
To appear in Advances in Geometry.)

## Tim

Tim Mussche is working on a thesis, among other things about the chromatic number
of the Kneser graph of a Grassmannian and that of the point-hyperplane flags.
Results on the former have perhaps never been written down and will have to be
reconstructed one day. The note below has some preliminary stuff on the latter.
Cocliques in the Kneser graph on point-hyperplane flags.

## Ralf

Ralf Gramlich wrote a thesis
("On Graphs, Geometries and Groups of Lie Type").
A preprint of that contained a characterization of graphs that are locally
the point-hyperplane antiflag graph of a projective space in dimension
at least 5. A prepreprint of a note that shows how to replace "at least 5"
by "at least 3" in that proof:
Graphs that are locally the
point-hyperplane antiflag graph of a projective space.

Ralf further simplified the proof, and the result can be seen in the final version
of his thesis, and was also published in

A.M. Cohen, F.G.M.T. Cuypers, R. Gramlich,
*Local recognition of non-incident point-hyperplane graphs*,
Combinatorica 25 (2005) 271-296.

## Sander

Sander van Rijnswou wrote a thesis
("Testing the equivalence of planar curves") that contained some discussion
on the invariants of the planar quartic. As it turns out, one can give all invariants:
Invariants of the ternary quartic.

(In the same way one also gets invariants of
binary forms.)

## Jochem

The M.Sc. thesis of Jochem Berndsen contains three parts.
The third part computes gossip numbers.
Here a slightly larger table.