The Combinatorial Structure of Wait-free Solvable Tasks (Extended Abstract)

S be the number of complete (m ?1)-simplexes in K m (colored with f0; : : :; m ? 1g), where the (m ? 1)-simplexes in each m-simplex are considered separately, and counted as +1 or ?1, by their induced orientations. We argue that S = I and S = C. To prove that S = I, consider the following cases. If an (m ? 1)-face is internal, then it contributes 0 to S, since the contributions of the two m-simplexes containing it cancel each other. Obviously, an internal (m ? 1)-face contributes 0 to I. An external (m ? 1)-face in the boundary of K m is counted the same, +1 or ?1 by orientation, in both S and I. Therefore, S = I. To prove that C = S, consider an m-simplex m , and look at the following cases. If m contains two (m?1)-faces which are completely colored, then m is not completely colored and contributes 0 to C. Note that m contributes 0 also to S, since the contributions of the two faces cancel each other. If m contains exactly one (m ? 1)-face which is completely colored (with f0; : : :; m ? 1g), then m must be completely colored and contributes +1 or ?1, by orientation, to C as well as to S. If m does not contain any (m ? 1)-face which is completely colored, then m is not completely colored and therefore, it contributes 0 to C as well as to S. Finally, note that m cannot contain more than two (m ? 1)-faces which are completely colored. We can now derive the oriented version of Sperner's Lemma.