Jon Folkman

 ( 1938 Births ) 

Jon Hal Folkman (December 8, 1938 – January 23, 1969)Birth and death dates from , and from , both of which were dedicated to the memory of Folkman. was an American mathematician, a student of John Milnor, and a researcher at the RAND Corporation.

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Schooling Folkman was a Putnam Fellow in 1960. Putnam competition results , Mathematical Association of America, retrieved 2010-10-17. He received his Ph.D. in 1964 from Princeton University, under the supervision of Milnor, with a thesis entitled Equivariant Maps of Spheres into the Classical Groups..

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Research Jon Folkman contributed important theorems in many areas of combinatorics.

In geometric combinatorics, Folkman is known for his pioneering and posthumously-published studies of ; in particular, the Folkman–Lawrence topological representation theorem. is "one of the cornerstones of the theory of oriented matroids".Page 17:

The Folkman-Lawrence representation theorem is called the "Lawrence representation theorem" by Günter M. Ziegler in remark 7.23 on page 211:

Günter M. Ziegler (1995). 038794365X, Springer-Verlag. 038794365X

In lattice theory, Folkman solved an open problem on the foundations of combinatorics by proving a conjecture of Gian-Carlo Rota; in proving Rota's conjecture, Folkman characterized the structure of the homology groups of matroid in terms of the free group of finite rank.

• (1986). 9780817631734, Birkhäuser Boston, Inc.. ISBN 9780817631734
• (1986). 9780817631734, Birkhäuser Boston, Inc.. ISBN 9780817631734
• (1986). 9780817631734, Birkhäuser Boston, Inc.. ISBN 9780817631734

In graph theory, he was the first to study semi-symmetric graphs, and he discovered the semi-symmetric graph with the fewest possible vertices, now known as the Folkman graph.. He proved the existence, for every positive h, of a finite K _h + 1-free graph which has a monocolored K_h in every 2-coloring of the edges, settling a problem previously posed by Paul Erdős and András Hajnal.. He further proved that if G is a finite graph such that every set S of vertices contains an independent set of size (| S| −  k)/2 then the chromatic number of G is at most k + 2.J. Folkman: An upper bound on the chromatic number of a graph, in: Combinatorial theory and its application, II (Proc. Colloq., Balatonfüred, 1969), North-Holland, Amsterdam, 1970, 437–457.

In convex geometry, Folkman worked with his RAND Corporation colleague Lloyd Shapley to prove the Shapley–Folkman lemma and theorem: Their results suggest that sums of sets are approximately convex; in mathematical economics their results are used to explain why economies with many agents have approximate equilibria, despite individual nonconvexities..

In additive combinatorics, Folkman's theorem states that for each assignment of finitely many colors to the positive integers, there exist arbitrarily large sets of integers all of whose nonempty sums have the same color; the name was chosen as a memorial to Folkman by his friends.Page 81 in . In Ramsey theory, the Rado–Folkman–Sanders theorem describes "partition regular" sets.

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The Folkman Number F(p, q; r) For r > max{p, q}, let F(p, q; r) denote the minimum number of vertices in a graph G that has the following properties:

1. G contains no complete subgraph on r vertices,
2. in any green-red coloring of the edges of G there is either a green K_p or a red K_q subgraph.

Some results are

• F(3, 3; 5) < 18 (Martin Erickson)
• F(2, 3; 4) < 1000 (Vojtěch Rödl, Andrzej Dudek)

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Brain cancer and despair In the late 1960s, Folkman suffered from brain cancer; while hospitalized, Folkman was visited repeatedly by Ronald Graham and Paul Erdős. After his brain surgery, Folkman was despairing that he had lost his mathematical skills. As soon as Folkman received Graham and Erdős at the hospital, Erdős challenged Folkman with mathematical problems, helping to rebuild his Self-confidence.

Folkman later purchased a gun and killed himself. Folkman's supervisor at RAND, Delbert Ray Fulkerson, blamed himself for failing to notice suicidal behaviors in Folkman. Several years later Fulkerson also killed himself..

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