In this note, we use the theory of Desarguesian spreads to investigate good eggs. Thas showed that an egg in $PG(4n-1,q)$, $q$ odd, with two good elements is elementary. By a short combinatorial argument, we show that a similar statement holds for large pseudo-caps, in odd and even characteristic. As a corollary, this improves and extends the result of Thas, Thas and Van Maldeghem (2006) where one needs at least $4$ good elements of an egg in even characteristic to obtain the same conclusion. We rephrase this corollary to obtain a characterisation of the generalised quadrangle $T_3(O)$ of Tits. Lavrauw (2005) characterises elementary eggs in odd characteristic as those good eggs containing a space that contains at least $5$ elements of the egg, but not the good element. We provide an adaptation of this characterisation for weak eggs in odd and even characteristic. As a corollary, we obtain a direct geometric proof for the theorem of Lavrauw.
@article{10_37236_4913,
author = {Sara Rottey and Geertrui Van de Voorde},
title = {Characterisations of elementary pseudo-caps and good eggs},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {1},
doi = {10.37236/4913},
zbl = {1311.51006},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4913/}
}
TY - JOUR
AU - Sara Rottey
AU - Geertrui Van de Voorde
TI - Characterisations of elementary pseudo-caps and good eggs
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/4913/
DO - 10.37236/4913
ID - 10_37236_4913
ER -
%0 Journal Article
%A Sara Rottey
%A Geertrui Van de Voorde
%T Characterisations of elementary pseudo-caps and good eggs
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/4913/
%R 10.37236/4913
%F 10_37236_4913
Sara Rottey; Geertrui Van de Voorde. Characterisations of elementary pseudo-caps and good eggs. The electronic journal of combinatorics, Tome 22 (2015) no. 1. doi: 10.37236/4913
