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The work is devoted to to one of the variations of the Hadwiger--Nelsonproblem on the chromatic number of the plane. In this formulation one needs tofind for arbitrarily small $\varepsilon$ the least possible number of colorsneeded to color the Euclidean plane in such a way that any two points, thedistance between which belongs to the interval $[1-\varepsilon,1+\varepsilon]$, are colored differently. The hypothesis proposed by G. Exoo in2004, states that for arbitrary positive $\varepsilon$ at least 7 colors arerequired. Also, with a sufficiently small $\varepsilon$ the number of colors isexactly 7. The main result of this paper is the proof of this hypothesis.

On the chromatic number of the plane with an arbitrarily short interval of forbidden distances