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The satisfiability problem is NP-complete but there are subclasses where allthe instances are satisfiable. For this, restrictions on the shape of theformula are made. Darman and D\"ocker show that the subclass MONOTONE$3$-SAT-($k$,1) with $k \geq 5$ proves to be NP-complete and pose the openquestion whether instances of MONOTONE $3$-SAT-(3,1) are satisfiable. Thispaper shows that all instances of MONOTONE $3$-SAT-(3,1) are satisfiable usingthe new concept of a color-structures.

All instances of MONOTONE 3-SAT-(3,1) are satisfiable