By a k k -graph we will mean a collection of k k -element subsets of some fixed set V V . A k k -graph can be regarded as a ( k − 1 ) (k - 1) -chain on 2 V {2^V} , the simplicial complex of all subsets of V V , over the coefficient group Z / 2 \mathbb {Z}/2 , the additive group of integers modulo 2 2 . The induced group structure on the ( k − 1 ) (k - 1) -chains leads to natural definitions of the coboundary δ \delta of a chain, the cochain complex of C = { C k , δ } C = \{ {C^k},\delta \} and the usual cohomology groups H k ( C ; Z / 2 ) {H^k}(C;\mathbb {Z}/2) . In particular, it is possible to construct what could be called "higher-order" coboundary operators δ ( i ) {\delta ^{(i)}} , where δ ( i ) {\delta ^{(i)}} increases dimension by i i (rather than just 1 1 ). In this paper we will develop various properties of these δ ( i ) {\delta ^{(i)}} , and in particular, compute the corresponding cohomology groups for 2 V {2^V} over Z / 2 \mathbb {Z}/2 . It turns out that these groups depend in a rather subtle way on the arithmetic properties of i i .