Directed Graph Ramsey Numbers
It is also possible to define Ramsey numbers for directed graphs. (These were introduced by P. Erdős & L. Moser.) Let R(n) be the smallest number Q such that any complete graph with singly directed arcs (also called a "tournament") and with ≥ Q nodes contains an acyclic (also called "transitive") n-node subtournament.
This is the directed-graph analogue of what (above) has been called R(n,n;2), the smallest number Z such that any 2-colouring of the edges of a complete undirected graph with ≥ 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.")
Indeed many find the directed graph problem to actually be more elegant than the unidirected one. We have R(0)=0, R(1)=1, R(2)=2, R(3)=4, R(4)=8, R(5)=14, R(6)=28, 32≤R(7)≤55, and R(8) is again a problem you do not want powerful aliens to pose.
Read more about this topic: Ramsey's Theorem
Famous quotes containing the words directed, graph and/or numbers:
“Having a thirteen-year-old in the family is like having a general-admission ticket to the movies, radio and TV. You get to understand that the glittering new arts of our civilization are directed to the teen-agers, and by their suffrage they stand or fall.”
—Max Lerner (b. 1902)
“In this Journal, my pen is a delicate needle point, tracing out a graph of temperament so as to show its daily fluctuations: grave and gay, up and down, lamentation and revelry, self-love and self-disgust. You get here all my thoughts and opinions, always irresponsible and often contradictory or mutually exclusive, all my moods and vapours, all the varying reactions to environment of this jelly which is I.”
—W.N.P. Barbellion (1889–1919)
“... there are persons who seem to have overcome obstacles and by character and perseverance to have risen to the top. But we have no record of the numbers of able persons who fall by the wayside, persons who, with enough encouragement and opportunity, might make great contributions.”
—Mary Barnett Gilson (1877–?)