  This paper is concerned with the large-scale regularity in the homogenization
of elliptic systems of elasticity with periodic high-contrast coefficients. We
obtain the large-scale Lipschitz estimate that is uniform with respect to the
contrast ratio $\delta^2$ for $0<\delta<\infty$. Our study also covers the case
of soft inclusions ($\delta=0$) as well as the case of stiff inclusions
($\delta=\infty$). The large-scale Lipschitz estimate, together with classical
local estimates, allows us to establish explicit bounds for the matrix of
fundamental solutions and its derivatives.
