For a Poisson process in R^2 with intensity 1, the distribution of the maximum number of points on an up/right path from (0,0) to (N,N) has been studied in detail, culminating in Baik-Deift-Johansson's celebrated Tracy-Widom fluctuation result. We consider a variant of the model where one adds, on the diagonal, some additional points according to a one dimensional Poisson process with rate...