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In this paper we present a quantitative analysis of the first and secondBorel-Cantelli Lemmas and of two of their generalisations: theErd\H{o}s-R\'enyi Theorem, and the Kochen-Stone Theorem. We will see that thefirst three results have direct quantitative formulations, giving an explicitrelationship between quantitative formulations of the assumptions and theconclusion. For the Kochen-Stone theorem, however, we can show that thenumerical bounds of a direct quantitative formulation are not computable ingeneral. Nonetheless, we obtain a quantitative formulation of the Kochen-StoneTheorem using Tao's notion of metastability.

On the Borel-Cantelli Lemmas, the Erd\H{o}s-R\'enyi Theorem, and the Kochen-Stone Theorem