The satisfiability problem is NP-complete but there are subclasses where all the instances are satisfiable. For this, restrictions on the shape of the formula 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 open question whether instances of MONOTONE $3$-SAT-(3,1) are satisfiable. This paper shows that all instances of MONOTONE $3$-SAT-(3,1) are satisfiable using the new concept of a color-structures.

翻译：可满足性问题是NP完全的，但存在所有实例均可满足的子类。为此，需对公式的形式施加限制。Darman与D\"ocker证明，当$k \geq 5$时，子类MONOTONE $3$-SAT-($k$,1)是NP完全的，并提出了MONOTONE $3$-SAT-(3,1)实例是否均可满足的开放问题。本文利用颜色结构（color-structures）这一新概念，证明所有MONOTONE $3$-SAT-(3,1)实例均可满足。