Ramsey's theorem and Poisson random measures (Q789817)

Let N be a random point process defined on a \(\delta\)-ring \({\mathcal D}\) of subsets of a measurable space and suppose that N has independent increments: whereas \(D_ 1,...,D_ k\in {\mathcal D}\) are disjoint the random variables \(N(D_ 1),...,N(D_ k)\) are independent. Define a set \(D\in {\mathcal D}\) to be small with respect to N if \(N(D)=0\) a.s. \textit{A. Prékopa} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 1, 153-170 (1958; Zbl 0089.340)] showed that if singletons belong to \({\mathcal D}\) and are small then necessarily N is a Poisson point process. The authors of this paper obtain the same conclusions under the formally weaker condition that for each \(D\in {\mathcal D}\) there exist a countable subfamily \({\mathcal B}\) of \({\mathcal D}\) such that \(D\subset U\{B:B\in {\mathcal B}\}\) and for each \(x\in D\), \(\cap \{B\in {\mathcal B}:x\in B\}\) is small. Their method of proof is based on appeal to Ramsey's theorem in combinatorial analysis.