  We study the problem of finding nonconstant monic integer polynomials,
normalized by their degree, with small supremum on an interval I. The monic
integer transfinite diameter t_M(I) is defined as the infimum of all such
supremums. We show that if I has length 1 then t_M(I) = 1/2.
  We make three general conjectures relating to the value of t_M(I) for
intervals I of length less that 4. We also conjecture a value for t_M([0, b])
where 0 < b < 1. We give some partial results, as well as computational
evidence, to support these conjectures.
  We define two functions that measure properties of the lengths of intervals I
with t_M(I) on either side of t. Upper and lower bounds are given for these
functions.
  We also consider the problem of determining t_M(I) when I is a Farey
interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning
this value is true for an infinite family of Farey intervals.
