Multiplicities of subgraphs

Abstract

A former conjecture of Burr and Rosta [l], extending a conjecture of Erd6s [2], asserted that in any two-colouring of the edges of a large complete graph, the proportion of subgraphs isomorphic to a fixed graph G which are monochromatic is at least the proportion found in a random eolouring. It is now known that the conjecture fails for some graphs G, including G = Kp for p>4.We investigate for which graphs G the conjecture holds. Our main result is that the conjecture fails if G contains K 4 as a subgraph, and in particular it fails for almost all graphs.

There exist graphs with super‐exponential Ramsey multiplicity constant

“…After it was shown for several graphs that C(G) = 2 1−e(G) , Burr and Rosta [2] made the stronger conjecture that C(G) = 2 1−e(G) holds for all graphs G. It is known that C(G) = 2 1−e(G) is true for forests, cycles, and wheels with an even number of spokes [2,4,5,10,[12][13][14][15].…”

Ramsey multiplicity of linear patterns in certain finite abelian groups

“…As before, a suitable version of such a statement is true for graphs (see Theorem 4 in [22]). We already noted in the introduction that the Cayley graph construction can be used to transfer a bad colouring from the arithmetic to the graph setting.…”

“…A relatively large number of linear systems can be shown to be common in this way. It is impossible to give an exhaustive list, but systems of equations associated with triangular edge-or vertex-trees, or those associated with square wheels and other regular grid structures (see Chapter II of [21]), fall in this category.…”

“…However, the Burr-Rosta conjecture has been verified for several classes of graphs, which we shall call common in keeping with the usual terminology. Graphs known to be common include trees, cycles, even-spoked wheels, triangular edge-and vertex-trees [29,30,22,36], to name just a few. Despite much research over the past two decades, the question of determining for which small graphs H the Burr-Rosta conjecture holds remains wide open.…”

The minimum number of monochromatic 4-term progressions in $\mbb{Z}_p$

“…The initial construction Thomason gave to disprove Erdős's conjecture was rather obscurely phrased in terms of quadratic forms over a finite field (compare with the quadratic nature of Gowers's construction in Section 3). Equivalent and much clearer formulations have since appeared in [8] and [11].…”