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5.M. SIX PEOPLE AT A PARTY -- RAMSEY THEORY
In a group of six people, there is a triple who all know each other or there is a triple who are all strangers. I.e., the Ramsey number R(3,3) = 6. I will not go into the more complex aspects of this -- see Graham & Spencer for a survey.
P. Erdös & G. Szekeres. A combinatorial problem in geometry. Compositio Math. 2 (1935) 463 470. [= Paul Erdös; The Art of Counting; Ed. by Joel Spencer, MIT Press, 1973, pp. 5 12.] They prove that if n  BC(a+b-2, a-1), then any two colouring of K_a contains a monochromatic K_a or K_b.

William Lowell Putnam Examination, 1953, part I, problem 2. In: L. E. Bush; The William Lowell Putnam Mathematical Competition; AMM 60 (1953) 539-542. Reprinted in: A. M. Gleason, R. E. Greenwood & L. M. Kelly; The William Lowell Putnam Mathematical Competition Problems and Solutions -- 1938 1964; MAA, 1980; pp. 38 & 365 366. The classic six people at a party problem.

R. E. Greenwood & A. M. Gleason. Combinatorial relations and chromatic graphs. Canadian J. Math. 7 (1955) 1-7. Considers n = n(a,b,...) such that a two colouring of K_n contains a K_a of the first colour or a K_b of the second colour or .... Thus n(3,3) = 6. They find the bound and many other results of Erdös & Szekeres.

C. W. Bostwick, proposer; John Rainwater & J. D. Baum, solvers. Problem E1321 -- A gathering of six people. AMM 65 (1958) 446 & 66 (1959) 141 142.

Gamow & Stern. 1958. Diagonal strings. Pp. 93 95.

G. J. Simmons. The Game of Sim. JRM 2 (1969) 66.

M. Gardner. SA (Jan 1973) c= Knotted, chap. 9. Exposits Sim. Reports Simmons' result that it is second person (determined after his 1969 article above). The Addendum in Knotted reports that several people have shown that Sim on five points is a draw. Numerous references.

Ronald L. Graham & Joel H. Spencer. Ramsey theory. SA 263:1 (Jul 1990) 80 85. Popular survey of Ramsey theory beginning from Ramsey and Erdös & Szekeres.