{"id":553,"date":"2009-02-03T13:23:13","date_gmt":"2009-02-03T17:23:13","guid":{"rendered":"http:\/\/michaelnielsen.org\/blog\/?p=553"},"modified":"2009-02-03T13:24:22","modified_gmt":"2009-02-03T17:24:22","slug":"the-polymath-project","status":"publish","type":"post","link":"https:\/\/michaelnielsen.org\/blog\/the-polymath-project\/","title":{"rendered":"The polymath project"},"content":{"rendered":"

Tim Gower’s experiment in massively collaborative mathematics is now underway. He’s dubbed it the “polymath project” – if you want to see posts related to the project, I suggest looking here<\/a>. <\/p>\n

The problem to be attacked can be understood (though probably not solved) with only a little undergraduate mathematics. It concerns a result known as the Density Hales-Jewett<\/a> theorem. This theorem asks us to consider the set [tex][ 3 ]^n[\/tex] of all length [tex]n[\/tex] strings over the alphabet [tex]1, 2, 3[\/tex]. So, for example, [tex]11321[\/tex] is in [tex][3]^5[\/tex]. The theorem concerns the existence of combinatorial lines<\/em> in subsets of [tex][3]^n[\/tex]. A combinatorial line is a set of three points in [tex][3]^n[\/tex], formed by taking a string with one or more wildcards in it, e.g., [tex]112*1**3\\ldots[\/tex], and replacing those wildcards by [tex]1[\/tex], [tex]2[\/tex] and [tex]3[\/tex], respectively. In the example I’ve given, the resulting combinatorial line is:<\/p>\n

[tex] \\{ 11211113\\ldots, 11221223\\ldots, 11231333\\ldots \\} [\/tex]<\/p>\n

The Density Hales-Jewett theorem asserts that for any [tex]\\delta > 0[\/tex], for sufficiently large [tex]n = n(\\delta)[\/tex], all subsets of [tex][3]^n[\/tex] of size at least [tex]\\delta 3^n[\/tex] contain a combinatorial line,<\/p>\n

Apparently, the original proof of the Density Hales-Jewett theorem used ergodic theory; Gowers’ challenge is to find a purely combinatorial proof of the theorem. More background can be found here<\/a>. Serious discussion of the problem starts here<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

Tim Gower’s experiment in massively collaborative mathematics is now underway. He’s dubbed it the “polymath project” – if you want to see posts related to the project, I suggest looking here. The problem to be attacked can be understood (though probably not solved) with only a little undergraduate mathematics. It concerns a result known as… Continue reading The polymath project<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62],"tags":[],"class_list":["post-553","post","type-post","status-publish","format-standard","hentry","category-polymath1","entry"],"_links":{"self":[{"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/posts\/553","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/comments?post=553"}],"version-history":[{"count":0,"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/posts\/553\/revisions"}],"wp:attachment":[{"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/media?parent=553"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/categories?post=553"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/michaelnielsen.org\/blog\/wp-json\/wp\/v2\/tags?post=553"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}