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Threshold functions for small subgraphs
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DOI：10.1017/s0305004100058655.png)
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In this note we shall study random labelled graphs. Denote bythe set of all graphs with n given labelled vertices andM(n)edges. As usual, we turn(M(n))into a probability space by giving all graphs the same probability. The question we address ourselves to is the following. Given a graphHand a constant p, 0 < p < 1, for what functionsM(n)is it true that the probability PM(n)(H ⊂ G) that a graphG∈(M(n))containsHtends topas n∞→? This question was posed by Erdös and Rényi (3), (4), who also proved several beautiful and surprising theorems. In order to state the main general result of Erdös and Rényi in this direction, and for our use later, we introduce some definitions.
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