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Branch of combinatorics with the philosophy that 'total disorder is impossible'. For example, Ramsey's theorem asserts that for each $n$, every sufficiently large graph either contains a clique of size $n$ or a stable set of size $n$.
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Where is the Erdős–Rado theorem stated in Erdős and Rado's Bull AMS paper?
It is also stated as Theorem 4.(i), I think, and again on pages 470 and 472 where the reference is given to earlier results.
For a recent easy-to-read presentation of the proof, see Theorem 5.1.4 in …