Erdős–Dushnik–Miller theorem

In the mathematical theory of infinite graphs, the Erdős–Dushnik–Miller theorem is a form of Ramsey's theorem stating that every infinite graph contains either a countably infinite independent set, or a clique with the same cardinality as the whole graph.[1]

The theorem was first published by Ben Dushnik and E. W. Miller (1941), in both the form stated above and an equivalent complementary form: every infinite graph contains either a countably infinite clique or an independent set with equal cardinality to the whole graph. In their paper, they credited Paul Erdős with assistance in its proof. They applied these results to the comparability graphs of partially ordered sets to show that each partial order contains either a countably infinite antichain or a chain of cardinality equal to the whole order, and that each partial order contains either a countably infinite chain or an antichain of cardinality equal to the whole order.[2]

The same theorem can also be stated as a result in set theory, using the arrow notation of Erdős & Rado (1956), as ${\displaystyle \kappa \rightarrow (\kappa ,\aleph _{0})^{2}}$. This means that, for every set ${\displaystyle S}$ of cardinality ${\displaystyle \kappa }$, and every partition of the ordered pairs of elements of ${\displaystyle S}$ into two subsets ${\displaystyle P_{1}}$ and ${\displaystyle P_{1}}$, there exists either a subset ${\displaystyle S_{1}\subset S}$ of cardinality ${\displaystyle \kappa }$ or a subset ${\displaystyle S_{2}\subset S}$ of cardinality ${\displaystyle \aleph _{0}}$, such that all pairs of elements of ${\displaystyle S_{i}}$ belong to ${\displaystyle P_{i}}$.[3] Here, ${\displaystyle P_{1}}$ can be interpreted as the edges of a graph having ${\displaystyle S}$ as its vertex set, in which ${\displaystyle S_{1}}$ (if it exists) is a clique of cardinality ${\displaystyle \kappa }$, and ${\displaystyle S_{2}}$ (if it exists) is a countably infinite independent set.[1]

If ${\displaystyle S}$ is taken to be the cardinal number ${\displaystyle \kappa }$ itself, the theorem can be formulated in terms of ordinal numbers with the notation ${\displaystyle \kappa \rightarrow (\kappa ,\omega )^{2}}$, meaning that ${\displaystyle S_{2}}$ (when it exists) has order type ${\displaystyle \omega }$. For uncountable regular cardinals ${\displaystyle \kappa }$ (and some other cardinals) this can be strengthened to ${\displaystyle \kappa \rightarrow (\kappa ,\omega +1)^{2}}$;[4] however, it is consistent that this strengthening does not hold for the cardinality of the continuum.[5]

The Erdős–Dushnik–Miller theorem has been called the first "unbalanced" generalization of Ramsey's theorem, and Paul Erdős's first significant result in set theory.[6]

References

1. Milner, E. C.; Pouzet, M. (1985), "The Erdős–Dushnik–Miller theorem for topological graphs and orders", Order, 1 (3): 249–257, doi:10.1007/BF00383601, MR 0779390, S2CID 123272176; see in particular Theorem 44
2. Dushnik, Ben; Miller, E. W. (1941), "Partially ordered sets", American Journal of Mathematics, 63 (3): 600–610, doi:10.2307/2371374, hdl:10338.dmlcz/100377, JSTOR 2371374, MR 0004862; see in particular Theorems 5.22 and 5.23
3. Erdős, Paul; Rado, R. (1956), "A partition calculus in set theory", Bull. Amer. Math. Soc., 62 (5): 427–489, doi:10.1090/S0002-9904-1956-10036-0, MR 0081864
4. Shelah, Saharon (2009), "The Erdős–Rado arrow for singular cardinals", Canadian Mathematical Bulletin, 52 (1): 127–131, doi:10.4153/CMB-2009-015-8, MR 2494318
5. Shelah, Saharon; Stanley, Lee J. (2000), "Filters, Cohen sets and consistent extensions of the Erdős–Dushnik–Miller theorem", The Journal of Symbolic Logic, 65 (1): 259–271, doi:10.2307/2586535, JSTOR 2586535, MR 1782118
6. Hajnal, András (1997), "Paul Erdős' set theory", The mathematics of Paul Erdős, II, Algorithms and Combinatorics, 14, Berlin: Springer, pp. 352–393, doi:10.1007/978-3-642-60406-5_33, MR 1425228; see in particular Section 3, "Infinite Ramsey theory – early papers", p. 353