99 research outputs found
Reducts of random hypergraphs
Cataloged from PDF version of article.For each k > 1, let rk be the countable universal homogeneous k-hypergraph. In this paper, we
shall classify the closed permutation groups G such that AUt(rk ) < G < &vn(rk). In particular,
we shall show that there exist only finitely many such groups G for each k 2 1. We shall also
show tha
Reducts of the random bipartite graph
Let be the random bipartite graph, a countable graph with two
infinite sides, edges randomly distributed between the sides, but no edges
within a side. In this paper, we investigate the reducts of that
preserve sides. We classify the closed permutation subgroups containing the
group , where is the group of all isomorphisms
and anti-isomorphisms of preserving the two sides. Our results rely on
a combinatorial theorem of Ne\v{s}et\v{r}il-R\"{o}dl and a strong finite
submodel property for
Permutations on the random permutation
The random permutation is the Fra\"iss\'e limit of the class of finite
structures with two linear orders. Answering a problem stated by Peter Cameron
in 2002, we use a recent Ramsey-theoretic technique to show that there exist
precisely 39 closed supergroups of the automorphism group of the random
permutation, and thereby expose all symmetries of this structure. Equivalently,
we classify all structures which have a first-order definition in the random
permutation.Comment: 18 page
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
Overgroups of the Automorphism Group of the Rado Graph
We are interested in overgroups of the automorphism group of the Rado graph.
One class of such overgroups is completely understood; this is the class of
reducts. In this article we tie recent work on various other natural
overgroups, in particular establishing group connections between them and the
reducts.Comment: 11 pages, 2 figure
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