{"id":23,"date":"2020-06-09T17:50:59","date_gmt":"2020-06-09T21:50:59","guid":{"rendered":"http:\/\/sites.nd.edu\/ramsey-theory-2020\/?p=23"},"modified":"2020-06-10T17:03:28","modified_gmt":"2020-06-10T21:03:28","slug":"blue-group-thursday-3-30-week-1","status":"publish","type":"post","link":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/2020\/06\/09\/blue-group-thursday-3-30-week-1\/","title":{"rendered":"Blue group &#8212; Thursday 3.30, week 1"},"content":{"rendered":"<p>Here&#8217;s the plan for Thursday for the <strong><span style=\"color: #0000ff\">Blue<\/span><\/strong> group:<\/p>\n<ul>\n<li><strong>Everyone<\/strong>: Read Section 1.3; do exercises 1.14 (easy), 1.18 and 1.20 (both moderate). You could also think about the following much harder variant of exercise 1.18: part of that exercise asks you to show that there is a sequence of reals of length \\(k^2\\) that has neither an increasing nor a decreasing subsequence of length \\(k+1\\). Show that this is best possible, that is, show that <em>every<\/em> sequence\u00a0of reals of length \\(k^2+1\\) has either an increasing or a decreasing subsequence of length \\(k+1\\).<\/li>\n<li><strong>Henry<\/strong>: present exercise 1.10.<\/li>\n<li><strong>Joe and Anthony:<\/strong> present these two exercises &#8212; show that if G is not connected, then its complement <em>is<\/em> connected, and show that a graph is bipartite if and only if it has no odd-length cycles.<\/li>\n<li><strong>Nick<\/strong>: present exercise 1.12.<\/li>\n<li><strong>Alex<\/strong>: present exercise 1.13.<\/li>\n<li><strong>Ryan<\/strong>: present proof of Theorem 1.16.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Here&#8217;s the plan for Thursday for the Blue group: Everyone: Read Section 1.3; do exercises 1.14 (easy), 1.18 and 1.20 (both moderate). You could also think about the following much harder variant of exercise 1.18: part of that exercise asks you to show that there is a sequence of reals of length that has neither &hellip; <a href=\"https:\/\/sites.nd.edu\/ramsey-theory-2020\/2020\/06\/09\/blue-group-thursday-3-30-week-1\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Blue group &#8212; Thursday 3.30, week 1<\/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-23","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\/23","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=23"}],"version-history":[{"count":3,"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/posts\/23\/revisions"}],"predecessor-version":[{"id":28,"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/posts\/23\/revisions\/28"}],"wp:attachment":[{"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/media?parent=23"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/categories?post=23"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.nd.edu\/ramsey-theory-2020\/wp-json\/wp\/v2\/tags?post=23"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}