Asymptotics of Chebyshev Polynomials, I. Subsets of $\mathbb{R}$

We consider Chebyshev polynomials, $T_n(z)$, for infinite, compact sets $\mathfrak{e}\subset \mathbb{R}$ (that is, the monic polynomials minimizing the sup-norm, $\mathrm{\parallel }{T}_n{\mathrm{\parallel }}_{\mathfrak{e}}$, on $\mathfrak{e}$). We resolve a $45+$ year old conjecture of Widom that for finite gap subsets of $\mathbb{R}$, his conjectured asymptotics (which we call Szeg\H{o}-Widom asymptotics) holds. We also prove the first upper bounds of the form $\Vert T_n \Vert_{\frak{e}} \leq Q C({\frak{e}})^n$(where$C(\frak{e})$isthelogarithmiccapacityof$\frak{e}$)foraclassof$\frak{e}$'s with an infinite number of components, explicitly for those $\mathfrak{e}\subset \mathbb{R}$ that obey a Parreau-Widom condition.