Prekopa's theorem gives a qualitative sufficient condition for a completely random point process to be Poisson. A generalization of this theorem is presented. The proof is elementary and uses a combinatorial principle known as Ramsey's theorem.
Timothy C. Brown. Joseph Kupka. "Ramsey's Theorem and Poisson Random Measures." Ann. Probab. 11 (4) 904 - 908, November, 1983. https://doi.org/10.1214/aop/1176993440