Quadruple systems with independent neighborhoods

A 4-graph is odd if its vertex set can be partitioned into two sets so that every edge intersects both parts in an odd number of points. Letb(n)=maxα{α(n−α3)+(n−α)(α3)}=(12+o(1))(n4) denote the maximum number of edges in an n-vertex odd 4-graph. Let n be sufficiently large, and let G be an n-vertex 4-graph such that for every triple xyz   of vertices, the neighborhood N(xyz)={w:wxyz∈G} is independent. We prove that the number of edges of G   is at most b(n)b(n). Equality holds only if G   is odd with the maximum number of edges. We also prove that there is ε>0ε>0 such that if the 4-graph G   has minimum degree at least (1/2−ε)(n3), then G is 2-colorable.Our results can be considered as a generalization of Mantel's theorem about triangle-free graphs, and we pose a conjecture about k-graphs for larger k as well.