19 research outputs found
Orthogonal polarity graphs and Sidon sets
Determining the maximum number of edges in an -vertex -free graph is
a well-studied problem that dates back to a paper of Erd\H{o}s from 1938. One
of the most important families of -free graphs are the Erd\H{o}s-R\'enyi
orthogonal polarity graphs. We show that the Cayley sum graph constructed using
a Bose-Chowla Sidon set is isomorphic to a large induced subgraph of the
Erd\H{o}s-R\'enyi orthogonal polarity graph. Using this isomorphism we prove
that the Petersen graph is a subgraph of every sufficiently large
Erd\H{o}s-R\'enyi orthogonal polarity graph.Comment: The authors would like to thank Jason Williford for noticing an error
in the proof of Theorem 1.2 in the previous version. This error has now been
correcte
The apparent structure of dense Sidon sets
The correspondence between perfect difference sets and transitive projective
planes is well-known. We observe that all known dense (i.e., close to
square-root size) Sidon subsets of abelian groups come from projective planes
through a similar construction. We classify the Sidon sets arising in this
manner from desarguesian planes and find essentially no new examples, but there
are many further examples arising from nondesarguesian planes. We conjecture
that all dense Sidon sets arise in this manner. We also give a brief bestiary
of somewhat smaller Sidon sets with a variety of algebraic origins, and for
some of them provide an overarching pattern.Comment: 16 page
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Multi-Sidon spaces over finite fields
Sidon spaces have been introduced by Bachoc, Serra and Z\'emor in 2017 in
connection with the linear analogue of Vosper's Theorem. In this paper, we
propose a generalization of this notion to sets of subspaces, which we call
multi-Sidon space. We analyze their structures, provide examples and introduce
a notion of equivalnce among them. Making use of these results, we study a
class of linear sets in PG determined by points and we
investigate multi-orbit cyclic subspace codes
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum