Neighborhood Variants of the KKM Lemma, Lebesgue Covering Theorem, and Sperner's Lemma on the Cube

We establish a “neighborhood” variant of the cubical KKM lemma and the Lebesgue covering theorem and deduce a discretized version which is a “neighborhood” variant of Sperner’s lemma on the cube. The main result is the following: for any coloring of the unit $d$-cube $[0,1{]}^d$ in which points on opposite faces must be given different colors, and for any $\epsilon >0$, there is an ${\mathrm{\ell }}_{\mathrm{\infty }}$ $\epsilon$-ball which contains points of at least $(1+\frac{\epsilon }{1+\epsilon }{)}^d$ different colors, (so in particular, at least $(1+\frac{2}{3}\epsilon {)}^d$ different colors for all sensible $\epsilon \in (0,\frac{1}{2}]$).