17 research outputs found
Graphs with few 3-cliques and 3-anticliques are 3-universal
For given integers k, l we ask whether every large graph with a sufficiently
small number of k-cliques and k-anticliques must contain an induced copy of
every l-vertex graph. Here we prove this claim for k=l=3 with a sharp bound. A
similar phenomenon is established as well for tournaments with k=l=4.Comment: 12 pages, 1 figur
Towards a characterisation of Sidorenko systems
A system of linear forms over is said
to be Sidorenko if the number of solutions to in any is asymptotically as at least the expected
number of solutions in a random set of the same density. Work of Saad and Wolf
(2017) and of Fox, Pham and Zhao (2019) fully characterises single equations
with this property and both sets of authors ask about a characterisation of
Sidorenko systems of equations.
In this paper, we make progress towards this goal. Firstly, we find a simple
necessary condition for a system to be Sidorenko, thus providing a rich family
of non-Sidorenko systems. In the opposite direction, we find a large family of
structured Sidorenko systems, by utilising the entropy method. We also make
significant progress towards a full classification of systems of two equations.Comment: 18 page
On tripartite common graphs
A graph H is common if the number of monochromatic copies of H in a
2-edge-colouring of the complete graph is minimised by the random colouring.
Burr and Rosta, extending a famous conjecture by Erdos, conjectured that every
graph is common. The conjectures by Erdos and by Burr and Rosta were disproved
by Thomason and by Sidorenko, respectively, in the late 1980s. Collecting new
examples for common graphs had not seen much progress since then, although very
recently, a few more graphs are verified to be common by the flag algebra
method or the recent progress on Sidorenko's conjecture.
Our contribution here is to give a new class of tripartite common graphs. The
first example class is so-called triangle-trees, which generalises two theorems
by Sidorenko and answers a question by Jagger, \v{S}\v{t}ov\'i\v{c}ek, and
Thomason from 1996. We also prove that, somewhat surprisingly, given any tree
T, there exists a triangle-tree such that the graph obtained by adding T as a
pendant tree is still common. Furthermore, we show that adding arbitrarily many
apex vertices to any connected bipartite graph on at most five vertices give a
common graph
Monochromatic triangles in three-coloured graphs
In 1959, Goodman determined the minimum number of monochromatic triangles in
a complete graph whose edge set is two-coloured. Goodman also raised the
question of proving analogous results for complete graphs whose edge sets are
coloured with more than two colours. In this paper, we determine the minimum
number of monochromatic triangles and the colourings which achieve this minimum
in a sufficiently large three-coloured complete graph.Comment: Some data needed to verify the proof can be found at
http://www.math.cmu.edu/users/jcumming/ckpsty
On uncommon systems of equations
A system of linear equations over is common if the number
of monochromatic solutions to in any two-colouring of is
asymptotically at least the expected number of monochromatic solutions in a
random two-colouring of . Motivated by existing results for
specific systems (such as Schur triples and arithmetic progressions), as well
as extensive research on common and Sidorenko graphs, the systematic study of
common systems of linear equations was recently initiated by Saad and Wolf.
Building upon earlier work of Cameron, Cilleruelo and Serra, as well as Saad
and Wolf, common linear equations have recently been fully characterised by
Fox, Pham and Zhao, who asked about common \emph{systems} of equations. In this
paper we move towards a classification of common systems of two or more linear
equations. In particular we prove that any system containing an arithmetic
progression of length four is uncommon, confirming a conjecture of Saad and
Wolf. This follows from a more general result which allows us to deduce the
uncommonness of a general system from certain properties of one- or
two-equation subsystems.Comment: 21 page