Uncommon Systems of Equations

  title={Uncommon Systems of Equations},
  author={Nina Kamcev and Anita Liebenau and Natasha Morrison},
A linear system L over Fq is common if the number of monochromatic solutions to L = 0 in any two-colouring of Fq is asymptotically at least the expected number of monochromatic solutions in a random two-colouring of Fq . 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… Expand
Linear configurations containing 4-term arithmetic progressions are uncommon
A linear configuration is said to be common in G if every 2-coloring of G yields at least the number of monochromatic instances of a randomly chosen coloring. Saad and Wolf asked whether, analogouslyExpand


Towards characterizing locally common graphs.
A graph H is common if the number of monochromatic copies of H in a 2-edge-coloring of the complete graph is asymptotically minimized by the random coloring. The classification of common graphs isExpand
The minimum number of monochromatic 4-term progressions in Z
In this paper we improve the lower bound given by Cameron, Cilleruelo and Serra for the minimum number of monochromatic 4term progressions contained in any 2-coloring of Zp with p a prime. We alsoExpand
On Schur Properties of Random Subsets of Integers
Abstract A classic result of I. Schur [9] asserts that for everyr⩾2 and fornsufficiently large, if the set [n]={1, 2, …, n} is partitioned intorclasses, then at least one of the classes contains aExpand
There exist graphs with super-exponential Ramsey multiplicity constant
  • J. Fox
  • Computer Science
  • J. Graph Theory
  • 2008
It is proved that for each positive integer E there is a graph G with E edges and C(G) ≤ E−E/2+o(E). Expand
Sidorenko's conjecture for blow-ups
A celebrated conjecture of Sidorenko and Erdős-Simonovits states that, for all bipartite graphs H, quasirandom graphs contain asymptotically the minimum number of copies of H taken over all graphsExpand
Non-Three-Colourable Common Graphs Exist
It is shown that the 5-wheel is common, the first example of a common graph that is not three-colourable and extension of a conjecture that every graph is common. Expand
On monochromatic solutions of equations in groups
We show that the number of monochromatic solutions of the equation x α1 1 x α2 2 ··· x αr r = g in a 2-coloring of a finite group G ,w here α1 ,...,α r are permutations and g ∈ G, depends only on theExpand
Graph products and monochromatic multiplicities
Arcane two-edge-colourings of complete graphs were described in [13], in which there are significantly fewer monochromaticKr's than in a random colouring (so disproving a conjecture of Erdős [2]).Expand
Multiplicities of subgraphs
It is shown that the conjecture that in any two-colouring of the edges of a large complete graph, the proportion of subgraphs isomorphic to a fixed graphG which are monochromatic is at least the proportion found in a random colouring fails, and in particular it fails for almost all graphs. Expand
Ramsey multiplicity of linear patterns in certain finite abelian groups
In this article we explore an arithmetic analogue of a long-standing open problem in graph theory: what can be said about the number of monochromatic additive configurations in 2-colourings of finiteExpand