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. Let b(n) = max α { α ( n− α 3 ) + (n− α) ( α 3 )} = ( 1 2 + o(1) )( n 4 ) 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). Equality holds only if G is odd with the maximum number of edges. We also prove that there is ε > 0 such that if a 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.