On the number of experiments sufficient and in the worst case necessary to identify all causal relations among N variables

We show that if any number of variables are allowed to be simultaneously and independently randomized in any one experiment, log_2(N) + 1 experiments are sufficient and in the worst case necessary to determine the causal relations among N ≥ 2 variables when no latent variables, no sample selection bias and no feedback cycles are present. For all K, 0 < K < 1/2 N we provide an upper bound on the number experiments required to determine causal structure when each experiment simultaneously randomizes K variables. For large N, these bounds are significantly lower than the N - 1 bound required when each experiment randomizes at most one variable. For k_(max) < N/2, we show that (N/k_(max) -1) + N/2k_(max) log_2(k_(max)) experiments are sufficient and in the worst case necessary. We offer a conjecture as to the minimal number of experiments that are in the worst case sufficient to identify all causal relations among N observed variables that are a subset of the vertices of a DAG