Small linear dependencies for binary vectors of low weight

We show that every set of m ' cn√n log log n vectors in {0, 1}n in which every vector has Hamming weight 3 contains a subset of O(log n) vectors that form a linear dependency. Our proof is based on showing that in every graph of average degree at least c log log n, every legal edge coloring produces a cycle in which one of the colors appears either once or twice. (In both results, c is some constant.) The results proved are used (in a companion work) in refutation algorithms for semirandom 3CNF formulas.