Improved lower bounds on the size of the smallest solution to a graph colouring problem, with an application to relation algebra

Let G = (V,E) be a graph and let C be a finite set of colours. An edge C-colouring of G is a function λ : E → C. If G is symmetric and λ(x, y) = λ(y, x), for all (x, y) ∈ E, then we say that λ is a symmetric edge C-colouring. Let n be a natural number and let Cn = {f, ci : i < n} be a set of n+1 colours. Using a probabilistic construction and an application of the Local Lemma we prove that there is a complete irreflexive graph Gn with at least two nodes and a symmetric edge Cn-colouring λn of Gn such that for any edge (x, y) of Gn and any β, γ ∈ Cn, f ∈ {λn(x, y), β, γ} ⇐⇒ ∃z ∈ Gn (β = λn(x, z) ∧ γ = λn(y, z)) Moreover, such a graph exists of size ( 3k−4 k ) provided k is large enough so that n ( 1− 1 n2 )(k−2)2 ( 1 + ( 2 ( 2k − 4 k ) + 2k ( 2k − 5 k − 1 ))2) ≤ 1 e (1) Equivalently, for k satisfying this inequality, the symmetric integral relation algebra with n + 1 diversity atoms one of which is flexible but where all inflexible diversity triangles are forbidden has a representation over a base of size ( 3k−4 k ) . This significantly reduces the size of the smallest known edge-labelled graph satisfying these conditions.