Monic integer Chebyshev problem

We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let ${\text{M}}_n(\mathbb{Z})$ denote the monic polynomials of degree $n$ with integer coefficients. A {\it monic integer Chebyshev polynomial} $M_n\in {\text{M}}_n(\mathbb{Z})$ satisfies $$\mathrm{\parallel }{M}_n{\mathrm{\parallel }}_E=\underset{P_n\in {\text{M}}_n(\mathbb{Z})}{\inf }\mathrm{\parallel }{P}_n{\mathrm{\parallel }}_E\mathrm{.}$$ and the {\it monic integer Chebyshev constant} is then defined by $$t_M(E):=\underset{n\to \mathrm{\infty }}{\lim }\mathrm{\parallel }{M}_n{\mathrm{\parallel }}_E^{1\mathrm{/}n}\mathrm{.}$$ This is the obvious analogue of the more usual {\it integer Chebyshev constant} that has been much studied. We compute $t_M(E)$ for various sets including all finite sets of rationals and make the following conjecture, which we prove in many cases. \medskip\noindent {\bf Conjecture.} {\it Suppose $[a_2\mathrm{/}{b}_2,a_1\mathrm{/}{b}_1]$ is an interval whose endpoints are consecutive Farey fractions. This is characterized by $a_1{b}_2-a_2{b}_1=1.$ Then} $$t_M[a_2\mathrm{/}{b}_2,a_1\mathrm{/}{b}_1]=\max (1\mathrm{/}{b}_1,1\mathrm{/}{b}_2)\mathrm{.}$$ This should be contrasted with the non-monic integer Chebyshev constant case where the only intervals where the constant is exactly computed are intervals of length 4 or greater.