Quasirandom permutations are characterized by 4-point densities

For permutations ${\pi}$ and ${\tau}$ of lengths ${|\pi|\le|\tau|}$ , let ${t(\pi,\tau)}$ be the probability that the restriction of ${\tau}$ to a random ${|\pi|}$ -point set is (order) isomorphic to ${\pi}$ . We show that every sequence ${\{\tau_j\}}$ of permutations such that ${|\tau_j|\to\infty}$ and ${t(\pi,\tau_j)\to 1/4!}$ for every 4-point permutation ${\pi}$ is quasirandom (that is, ${t(\pi,\tau_j)\to 1/|\pi|!}$ for every ${\pi}$ ). This answers a question posed by Graham.