On Jumping Densities of Hypergraphs

Let r ≥ 2 be an integer. The real number α ∈ [0, 1] is a jump for rif there exists c > 0 such that for every positive and every integer m ≥ r, every r-uniform graph with n > n 0 (, m) vertices andat least (α +) n r edges contains asubgraph with m vertices and at least (α + c) m r edges. A fundamental result in extremal graph theory due to Erd˝ os and Stone implies that every number in [0, 1) is a jump for r = 2. Erd˝ os also showed that every number in [0, r!/r r) is a jump for r ≥ 3. However, not every number in [0, 1) is a jump for r ≥ 3. In fact, Frankl and Rödl showed the existence of non-jumps for r ≥ 3. In this talk, we describe more recent results for r ≥ 3.