Berarducci, Alessandro; Majer, Pietro; Novaga, Matteo Infinite paths and cliques in random graphs. (English) Zbl 1243.05216 Fundam. Math. 216, No. 2, 163-191 (2012). Summary: We study the thresholds for the emergence of various properties in random subgraphs of \((\mathbb N, <)\). In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory. MSC: 05C80 Random graphs (graph-theoretic aspects) 05C55 Generalized Ramsey theory 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 60C05 Combinatorial probability 06A07 Combinatorics of partially ordered sets Keywords:random graphs; Ramsey theory; percolation threshold; probability PDFBibTeX XMLCite \textit{A. Berarducci} et al., Fundam. Math. 216, No. 2, 163--191 (2012; Zbl 1243.05216) Full Text: DOI arXiv