  We derive exact relations that connect the universal $C/k^4$-decay of the
momentum distribution at large $k$ with both thermodynamic properties and
correlation functions of two-component Fermi gases in one dimension with
contact interactions. The relations are analogous to those obtained by Tan in
the three-dimensional case and are derived from an operator product expansion
of the one- and two-particle density matrix. They extend earlier results by
Olshanii and Dunjko [Phys. Rev. Lett. 91, 090401 (2003)] for the bosonic
Lieb-Liniger gas. As an application, we calculate the pair distribution
function at short distances and the dimensionless contact in the limit of
infinite repulsion. The ground state energy approaches a universal constant in
this limit, a behavior that also holds in the three-dimensional case. In both
one and three dimensions, a Stoner instability to a saturated ferromagnet for
repulsive fermions with zero range interactions is ruled out at any finite
coupling.
