{"id":59,"date":"2020-06-26T17:06:58","date_gmt":"2020-06-26T21:06:58","guid":{"rendered":"http:\/\/sites.nd.edu\/ramsey-theory-2020\/?p=59"},"modified":"2020-06-26T17:06:58","modified_gmt":"2020-06-26T21:06:58","slug":"tuesday-of-week-4-red-group","status":"publish","type":"post","link":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/2020\/06\/26\/tuesday-of-week-4-red-group\/","title":{"rendered":"Tuesday of week 4 – red group"},"content":{"rendered":"

For Tuesday, everyone should read up through the end of Section 3.1, and do all the exercises. In general when reading mathematics it is a good idea to have paper to hand, to draw pictures and verify claims. This section is one of those situations where it is not a good idea — it is absolutely essential<\/em>.<\/p>\n

At the end of last time, we saw the “AP focussing” argument used to show that for each \\(k\\), \\(W(k,2)\\) exists, assuming that \\(W(k-1,r)\\) exists for every \\(r\\). Next, we will move onto to the argument that \\(W(3,3)\\) is finite. This requires a new ingredient, because now we need to focus three<\/strong> monochromatic APs of length 2 (all of different colors) on a single, final focal point. Casey<\/strong> can present this.<\/p>\n

The argument that \\(W(3,3)\\) is finite generalizes to show that \\(W(3,r)\\) is finite, for every \\(r\\). Ted<\/strong> can present this.<\/p>\n

The last special case we’ll consider is \\(W(4,3)\\) (requiring focussing three monochromatic APs of length 3, all of different colors, on a single, final focal point). Colin<\/strong>\u00a0can present this.<\/p>\n

We should now be in a position to understand what’s going on in the proof of the general statement, that for each particular choice of \\(k\\) and \\(r\\), \\(W(k,r)\\) is finite (under the inductive hypothesis that \\(W(k-1,r’)\\) is finite for every<\/strong> \\(r’\\)). Bailee<\/strong> can present this.<\/p>\n","protected":false},"excerpt":{"rendered":"

For Tuesday, everyone should read up through the end of Section 3.1, and do all the exercises. In general when reading mathematics it is a good idea to have paper to hand, to draw pictures and verify claims. This section is one of those situations where it is not a good idea — it is … Continue reading Tuesday of week 4 – red group<\/span><\/a><\/p>\n","protected":false},"author":3751,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-59","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/posts\/59","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/users\/3751"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/comments?post=59"}],"version-history":[{"count":1,"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/posts\/59\/revisions"}],"predecessor-version":[{"id":60,"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/posts\/59\/revisions\/60"}],"wp:attachment":[{"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/media?parent=59"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/categories?post=59"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/tags?post=59"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}