In reverse mathematics, there is a significant difference in proof strength between the version of Ramsey's theorem for infinite graphs (the case "n" = 2) and for infinite multigraphs (the case "n" ≥ 3). The multigraph version of the theorem is equivalent in strength to the arithmetical comprehension axiom, making it part of the subsystem ACA0 of second-order arithmetic, one of the big five subsystems in reverse mathematics. In contrast, by a theorem of David Seetapun, the graph version of the theorem is weaker than ACA0, and (combining Seetapun's result with others) it does not fall into one of the big five subsystems. It is possible to deduce the finite Ramsey theorem from the infinite version by a proof by contradiction. Suppose the finite Ramsey theorem is false. Then there exist integers "c", "n", "T" such that for every integer "k", there exists a "c"-colouring of formula_26 without a monochromatic set of size "T". Let "C""k" denote the "c"-colourings of formula_26 without a monochromatic set of size "T". For any "k", the restriction of a colouring in "C""k"+1 to formula_26 (by ignoring the colour of all sets containing "k" + 1) is a colouring in "C""k". Define formula_29 to be the colourings in "C""k" which are restrictions of colourings in "C""k"+1. Since "C""k"+1 is not empty, neither is formula_30. Similarly, the restriction of any colouring in formula_31 is in formula_29, allowing one to define formula_33 as the set of all such restrictions, a non-empty set. Continuing so, define formula_34 for all integers "m", "k". Now, for any integer "k", formula_35, and each set is non-empty. Furthermore, "Ck" is finite as formula_36. It follows that the intersection of all of these sets is non-empty, and let formula_37. Then every colouring in "D""k" is the restriction of a colouring in "D""k"+1. Therefore, by unrestricting a colouring in "D""k" to a colouring in "D""k"+1, and continuing doing so, one constructs a colouring of formula_38 without any monochromatic set of size "T".