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The late nineteenth century saw more alleged solutions of the four-color problems, many of which stood for as long as 11 years Eventually errors were found, and the problem remained open on into the twentieth century What is particularly striking is that Gerhard Ringel (1919 2008) and J W T Youngs (1910 1970) were able to prove in 1968 that all of Heawood s estimates, for the chromatic number of any surface of genus at least 1, are sharp So the chromatic
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number of a torus is indeed 7 The chromatic number of a double-torus with two holes is 8 And so forth But the Ringel/Youngs proof, just like the Heawood formula, does not apply to the sphere They could not improve on Heawood s result that ve colors will always suf ce The four-color problem remained unsolved Then in 1974 there was blockbuster news Using 1200 hours of computer time on the University of Illinois supercomputer, Kenneth Appel and Wolfgang Haken showed that in fact four colors will always work to color any map on the sphere Their technique is to identify 633 fundamental con gurations of maps (to which all others can be reduced) and to prove that each of them is reducible to a simpler con guration But the number of fundamental con gurations was very large, and the number of reductions required was beyond the ability of any human to count And the reasoning is extremely intricate and complicated Enter the computer In those days computing time was expensive and not readily available, and Appel and Haken certainly could not get a 1200-hour contiguous time slice for their work So the calculations were done late at night, off the record, during various down times In fact, Appel and Haken did not know for certain whether the calculation would ever cease Their point of view was this: 1 If the computer nally stopped then it will have checked all the cases and the four-color problem was solved 2 If the computer never stopped then they could draw no conclusion Well, the computer stopped But the level of discussion and gossip and disagreement in the mathematical community did not Was this really a proof The computer had performed tens of millions of calculations Nobody could ever check them all But now the plot thickens Because in 1975 a mistake was found in the proof Speci cally, there was something amiss with the algorithm that Appel and Haken fed into the computer It was later repaired The paper was published in 1976 The four-color problem was declared to be solved In a 1986 article, Appel and Haken point out that the reader of their seminal 1976 article must face 1 50 pages containing text and diagrams; 2 85 pages lled with almost 2500 additional diagrams; 3 400 micro che pages that contain further diagrams and thousands of individual veri cations of claims made in the 24 statements in the main section of the text But it seems as though there is always trouble in paradise Errors continued to be discovered in the Appel/Haken proof Invariably the errors were xed But the stream of errors never seemed to cease So is the Appel/Haken work really a proof
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Well, there is hardly anything more reassuring than another, independent proof Paul Seymour and his group at Princeton University found another way to attack the problem In fact they found a new algorithm that seems to be more stable They also needed to rely on computer assistance But by the time they did their work computers were much, much faster So they required much less computer time In any event, this paper appeared in 1994
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