We consider the problem of partitioning n agents in an undirected social network into k almost equal size (differing by at most one) groups, where utility agent for a group is number her neighbors group. The core and envy-freeness are two compelling axiomatic fairness guarantees such settings. former demands that there be no coalition each has more than own group, while latter envy another they...

An ordered r-graph is an r-uniform hypergraph whose vertex set linearly ordered. Given 2?k?r, H interval k-partite if there exist at least k disjoint intervals in the ordering such that every edge of has nonempty intersection with each and contained their union. Our main result implies ?>k?1, then for d>0 n-vertex dn? edges some m?n m-vertex subgraph ?(dm?) edges. This extension to r-graphs obs...