Editing Ramsey's theorem (section)

=== Directed graphs ===
It is also possible to define Ramsey numbers for ''directed'' graphs; these were introduced by {{harvs|first1=P.|last1=Erdős|author1-link=Paul Erdős|first2=L.|last2=Moser|year=1964|txt}}. Let {{math|''R''(''n'')}} be the smallest number {{mvar|Q}} such that any complete graph with singly directed arcs (also called a "tournament") and with {{math|≥ ''Q''}} nodes contains an acyclic (also called "transitive") {{mvar|n}}-node subtournament.

This is the directed-graph analogue of what (above) has been called {{math|''R''(''n'', ''n''; 2)}}, the smallest number {{mvar|Z}} such that any 2-colouring of the edges of a complete ''un''directed graph with {{math|≥ ''Z''}} nodes, contains a monochromatic complete graph on n nodes. (The directed analogue of the two possible arc ''colours'' is the two ''directions'' of the arcs, the analogue of "monochromatic" is "all arc-arrows point the same way"; i.e., "acyclic.")

We have {{math|1=''R''(0) = 0}}, {{math|1=''R''(1) = 1}}, {{math|1=''R''(2) = 2}}, {{math|1=''R''(3) = 4}}, {{math|1=''R''(4) = 8}}, {{math|1=''R''(5) = 14}}, {{math|1=''R''(6) = 28}}, and {{math|34 ≤ ''R''(7) ≤ 47}}.<ref>{{citation|last1=Smith|first1=Warren D.|title=Partial Answer to Puzzle #27: A Ramsey-like quantity|url=http://rangevoting.org/PuzzRamsey.html|access-date=2020-06-02|last2=Exoo|first2=Geoff}}</ref><ref>{{cite arXiv|last1=Neiman|first1=David|last2=Mackey|first2=John|last3=Heule|first3=Marijn|date=2020-11-01|title=Tighter Bounds on Directed Ramsey Number R(7)|class=math.CO|eprint=2011.00683}}</ref>