In this work we analyze a direct product test in which each of two provers receives a subset of size n of a ground set U, and the two subsets intersect in about (1-\delta)n elements. We show that if each of the provers provides labels to the n elements it received, and the labels of the two provers agree in the intersection between the subsets with non-negligible probability, then the answers of the provers must correspond to a certain global assignment to the elements of U.

While previous results only worked for intersection of size at most n/2, in our model the questions and expected answers of the two provers are nearly identical. This is related to a recent construction of a unique games instance (ECCC TR14-142) where this setup arises at the ``outer verifier'' level.

Our main tool is a hypercontractive bound on the Bernoulli-Laplace model (aka a slice of the Boolean hypercube), from which we can deduce a ``small set expansion''-type lemma. We then use ideas from a recent work of the author about ``fortification'' to reduce the case of large intersection to the already studied case of smaller intersection.