Representations of families of triples over GF(2)

AbstractLet B be any family of 3-subsets of [n] = {1, …, n} such that every i in [n] belongs to at most three members of B. It is shown here that there exists a 3 × n (0, 1)-matrix M such that every set of columns of M indexed by a member of B is linearly independent over GF(2). The proof depends on finding a suitable vertex-coloring for the associated 3-uniform hypergraph. This matrix result, which is a special case of a conjecture of Griggs and Walker, implies the corresponding special case of a conjecture of Chung, Frankl, Graham, and Shearer and of Faudree, Schelp, and Sós concerning intersecting families of subsets

Triangle-Intersecting Families of Graphs

A family of graphs F is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. A conjecture of Simonovits and Sos from 1976 states that the largest triangle-intersecting families of graphs on a fixed set of n vertices are those obtained by fixing a specific triangle and taking all graphs containing it, resulting in a family of size (1/8) 2^{n choose 2}. We prove this conjecture and some generalizations (for example, we prove that the same is true of odd-cycle-intersecting families, and we obtain best possible bounds on the size of the family under different, not necessarily uniform, measures). We also obtain stability results, showing that almost-largest triangle-intersecting families have approximately the same structure.Comment: 43 page