Splitting property via shadow systems

Let M_k^r denote the set of r-element multisets over the set {1,...,k. We show that M_k^k has the so-called splitting property introduced by Ahlswede et al. Our approach gives a new interpretation of Sidorenko's construction and is applicable to give an upper bound on weighted Turán numbers, matching previous bounds. We also show how these results are connected to Tuza's conjecture on minimum triangle covers.