For a given set M of positive integers, Motzkin asked to find the
quantity μ(M), which is the supremum
of all upper densities δ(S),
where S is a set of nonnegative integers with the property that
a ∈ S, b ∈ S
⇒ a - b ∉ M. This problem is equivalent to two
colouring problems in graph theory. The first problem
is the "Asymptotic T-colouring efficiency"
due to Rabinowitz and Proulx and the second problem is the
"Fractional chromatic number of the distance graph generated by M"
due to Chang, Liu and Zhu. In this
seminar I will survey the progress in the density problem and show the
equivalence of the density problem
with the colouring problems.