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Borsuk's conjecture was believed to be true for 60 years till its counterexample was found in 1993 by Jeff Kahn and Gil Kalai.

They constructed an infinite family of counterexamples by using a result of Frankl and Wilson: https://www.ams.org/journals/bull/1993-29-01/S0273-0979-1993-00398-7/S0273-0979-1993-00398-7.pdf

Here's what Babai and Frankl say about the demise of Borsuk's conjecture in their manuscript "Linear Algebra Methods in Combinatorics":

Dead at the age of 60. Died after no apparent signs of illness, unexpectedly, of grave combinatorial causes. The news of the demise of Borsuk's venerable conjecture (1933) spread like brushfire among combinatorialists in Summer 1992. The disproof, found by Jeff Kahn (Rutgers) and Gil Kalai (Hebrew University), was the hot topic between lectures at conferences (the result came too late to be included on the regular programs). Countless copies of the manuscript traveled over electronic networks, silently crossing oceans and continents at lightening speed. The authors of this book found out about the result in more conventional ways. One of us heard it from Kahn himself while examining Gabi Bollobás's remarkable sculptures at the reception at a meeting in Cambridge, England. By then, in Tokyo, the other author had learned about it in a telephone conversation with a friend in New Jersey.

What Kahn communicated in a few minutes and without the benefit of paper or blackboard was not just the news of the result but also the complete proof. Remarkably, Borsuk's geometric conjecture was disproved in just a few lines, relying on the Frankl-Wilson Theorem (Theorem 7.15), a modular version of the RW theorem.

Also see this very recent survey article by Gil Kalai: https://arxiv.org/pdf/1505.04952v1.pdf

Hirsch Conjecture is another possible example. In 2010, Francisco Santos constructed a 43-dimensional polytope of 86 facets with a diameter of more than 43: https://doi.org/10.4007/annals.2012.176.1.7.

Also see this survey by Ziegler: http://www.math.uiuc.edu/documenta/vol-ismp/22_ziegler-guenter.pdf

Borsuk's conjecture was believed to be true for 60 years till its counterexample was found in 1993 by Jeff Kahn and Gil Kalai.

They constructed an infinite family of counterexamples by using a result of Frankl and Wilson: https://www.ams.org/journals/bull/1993-29-01/S0273-0979-1993-00398-7/S0273-0979-1993-00398-7.pdf

Here's what Babai and Frankl say about the demise of Borsuk's conjecture in their manuscript "Linear Algebra Methods in Combinatorics":

Dead at the age of 60. Died after no apparent signs of illness, unexpectedly, of grave combinatorial causes. The news of the demise of Borsuk's venerable conjecture (1933) spread like brushfire among combinatorialists in Summer 1992. The disproof, found by Jeff Kahn (Rutgers) and Gil Kalai (Hebrew University), was the hot topic between lectures at conferences (the result came too late to be included on the regular programs). Countless copies of the manuscript traveled over electronic networks, silently crossing oceans and continents at lightening speed. The authors of this book found out about the result in more conventional ways. One of us heard it from Kahn himself while examining Gabi Bollobás's remarkable sculptures at the reception at a meeting in Cambridge, England. By then, in Tokyo, the other author had learned about it in a telephone conversation with a friend in New Jersey.

What Kahn communicated in a few minutes and without the benefit of paper or blackboard was not just the news of the result but also the complete proof. Remarkably, Borsuk's geometric conjecture was disproved in just a few lines, relying on the Frankl-Wilson Theorem (Theorem 7.15), a modular version of the RW theorem.

Also see this very recent survey article by Gil Kalai: https://arxiv.org/pdf/1505.04952v1.pdf

Hirsch Conjecture is another possible example. In 2010, Francisco Santos constructed a 43-dimensional polytope of 86 facets with a diameter of more than 43: https://doi.org/10.4007/annals.2012.176.1.7.

Also see this survey by Ziegler: http://www.math.uiuc.edu/documenta/vol-ismp/22_ziegler-guenter.pdf (Wayback Machine)

Borsuk's conjecture was believed to be true for 60 years till its counterexample was found in 1993 by Jeff Kahn and Gil Kalai.

They constructed an infinite family of counterexamples by using a result of Frankl and Wilson: http://www.ams.org/journals/bull/1993-29-01/S0273-0979-1993-00398-7/S0273-0979-1993-00398-7.pdf

Here's what Babai and Frankl say about the demise of Borsuk's conjecture in their manuscript "Linear Algebra Methods in Combinatorics":

Dead at the age of 60. Died after no apparent signs of illness, unexpectedly, of grave combinatorial causes. The news of the demise of Borsuk's venerable conjecture (1933) spread like brushfire among combinatorialists in Summer 1992. The disproof, found by Jeff Kahn (Rutgers) and Gil Kalai (Hebrew University), was the hot topic between lectures at conferences (the result came too late to be included on the regular programs). Countless copies of the manuscript traveled over electronic networks, silently crossing oceans and continents at lightening speed. The authors of this book found out about the result in more conventional ways. One of us heard it from Kahn himself while examining Gabi Bollobás's remarkable sculptures at the reception at a meeting in Cambridge, England. By then, in Tokyo, the other author had learned about it in a telephone conversation with a friend in New Jersey.

What Kahn communicated in a few minutes and without the benefit of paper or blackboard was not just the news of the result but also the complete proof. Remarkably, Borsuk's geometric conjecture was disproved in just a few lines, relying on the Frankl-Wilson Theorem (Theorem 7.15), a modular version of the RW theorem.

Also see this very recent survey article by Gil Kalai: http://arxiv.org/pdf/1505.04952v1.pdf

Hirsch Conjecture is another possible example. In 2010, Francisco Santos constructed a 43-dimensional polytope of 86 facets with a diameter of more than 43: http://annals.math.princeton.edu/2012/176-1/p07.

Also see this survey by Ziegler: http://www.math.uiuc.edu/documenta/vol-ismp/22_ziegler-guenter.pdf

Borsuk's conjecture was believed to be true for 60 years till its counterexample was found in 1993 by Jeff Kahn and Gil Kalai.

They constructed an infinite family of counterexamples by using a result of Frankl and Wilson: https://www.ams.org/journals/bull/1993-29-01/S0273-0979-1993-00398-7/S0273-0979-1993-00398-7.pdf

Here's what Babai and Frankl say about the demise of Borsuk's conjecture in their manuscript "Linear Algebra Methods in Combinatorics":

Dead at the age of 60. Died after no apparent signs of illness, unexpectedly, of grave combinatorial causes. The news of the demise of Borsuk's venerable conjecture (1933) spread like brushfire among combinatorialists in Summer 1992. The disproof, found by Jeff Kahn (Rutgers) and Gil Kalai (Hebrew University), was the hot topic between lectures at conferences (the result came too late to be included on the regular programs). Countless copies of the manuscript traveled over electronic networks, silently crossing oceans and continents at lightening speed. The authors of this book found out about the result in more conventional ways. One of us heard it from Kahn himself while examining Gabi Bollobás's remarkable sculptures at the reception at a meeting in Cambridge, England. By then, in Tokyo, the other author had learned about it in a telephone conversation with a friend in New Jersey.

What Kahn communicated in a few minutes and without the benefit of paper or blackboard was not just the news of the result but also the complete proof. Remarkably, Borsuk's geometric conjecture was disproved in just a few lines, relying on the Frankl-Wilson Theorem (Theorem 7.15), a modular version of the RW theorem.

Also see this very recent survey article by Gil Kalai: https://arxiv.org/pdf/1505.04952v1.pdf

Hirsch Conjecture is another possible example. In 2010, Francisco Santos constructed a 43-dimensional polytope of 86 facets with a diameter of more than 43: https://doi.org/10.4007/annals.2012.176.1.7.

Also see this survey by Ziegler: http://www.math.uiuc.edu/documenta/vol-ismp/22_ziegler-guenter.pdf

Borsuk's conjecture was believed to be true for 60 years till its counterexample was found in 1993 by Jeff Kahn and Gil Kalai.

They constructed an infinite family of counterexamples by using a result of Frankl and Wilson: http://www.ams.org/journals/bull/1993-29-01/S0273-0979-1993-00398-7/S0273-0979-1993-00398-7.pdf

Here's what Babai and Frankl say about the demise of Borsuk's conjecture in their manuscript "Linear Algebra Methods in Combinatorics":

Dead at the age of 60. Died after no apparent signs of illness, unexpectedly, of grave combinatorial causes. The news of the demise of Borsuk's venerable conjecture (1933) spread like brushfire among combinatorialists in Summer 1992. The disproof, found by Jeff Kahn (Rutgers) and Gil Kalai (Hebrew University), was the hot topic between lectures at conferences (the result came too late to be included on the regular programs). Countless copies of the manuscript traveled over electronic networks, silently crossing oceans and continents at lightening speed. The authors of this book found out about the result in more conventional ways. One of us heard it from Kahn himself while examining Gabi Bollobás's remarkable sculptures at the reception at a meeting in Cambridge, England. By then, in Tokyo, the other author had learned about it in a telephone conversation with a friend in New Jersey.

 

What Kahn communicated in a few minutes and without the benefit of paper or blackboard was not just the news of the result but also the complete proof. Remarkably, Borsuk's geometric conjecture was disproved in just a few lines, relying on the Frankl-Wilson Theorem (Theorem 7.15), a modular version of the RW theorem.

Also see this very recent survey article by Gil Kalai: http://arxiv.org/pdf/1505.04952v1.pdf

Hirsch Conjecture is another possible example. In 2010, Francisco Santos constructed a 43-dimensional polytope of 86 facets with a diameter of more than 43: http://annals.math.princeton.edu/2012/176-1/p07.

Also see this survey by Ziegler: http://www.math.uiuc.edu/documenta/vol-ismp/22_ziegler-guenter.pdf

Borsuk's conjecture was believed to be true for 60 years till its counterexample was found in 1993 by Jeff Kahn and Gil Kalai.

They constructed an infinite family of counterexamples by using a result of Frankl and Wilson: http://www.ams.org/journals/bull/1993-29-01/S0273-0979-1993-00398-7/S0273-0979-1993-00398-7.pdf

Here's what Babai and Frankl say about the demise of Borsuk's conjecture in their manuscript "Linear Algebra Methods in Combinatorics":

Dead at the age of 60. Died after no apparent signs of illness, unexpectedly, of grave combinatorial causes. The news of the demise of Borsuk's venerable conjecture (1933) spread like brushfire among combinatorialists in Summer 1992. The disproof, found by Jeff Kahn (Rutgers) and Gil Kalai (Hebrew University), was the hot topic between lectures at conferences (the result came too late to be included on the regular programs). Countless copies of the manuscript traveled over electronic networks, silently crossing oceans and continents at lightening speed. The authors of this book found out about the result in more conventional ways. One of us heard it from Kahn himself while examining Gabi Bollobás's remarkable sculptures at the reception at a meeting in Cambridge, England. By then, in Tokyo, the other author had learned about it in a telephone conversation with a friend in New Jersey.

What Kahn communicated in a few minutes and without the benefit of paper or blackboard was not just the news of the result but also the complete proof. Remarkably, Borsuk's geometric conjecture was disproved in just a few lines, relying on the Frankl-Wilson Theorem (Theorem 7.15), a modular version of the RW theorem.

Also see this very recent survey article by Gil Kalai: http://arxiv.org/pdf/1505.04952v1.pdf

Hirsch Conjecture is another possible example. In 2010, Francisco Santos constructed a 43-dimensional polytope of 86 facets with a diameter of more than 43: http://annals.math.princeton.edu/2012/176-1/p07.

Also see this survey by Ziegler: http://www.math.uiuc.edu/documenta/vol-ismp/22_ziegler-guenter.pdf