Let $X$ be a smooth projective surface and $B$ a divisor with normal crossings such that $L$ = $K_X$ + $B$ is big. What is the smallest possible volume of $L$? The answer to this and related questions has many applications to compact moduli of surfaces of general type and to automorphisms of surfaces. In a joint work with Wenfei Liu, we improve several existing records. The new examples are based on a certain combinatorial problem which I will explain.