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Show that for any finite <math>r</math> there is a finite <math>N</math> such that for all <math>n\\ge N</math>, for any <math>r</math>-coloring of non-empty subsets of <math>[n]</math>, there always exist <math>1\\le i<j<k\\le n</math> such that the intervals <math>[i,j)=\\{i,i+1,\\ldots, j-1\\}</math>, <math>[j,k)=\\{j,j+1,\\ldots, k-1\\}<...\""},{"logid":1528,"ns":0,"title":"\u8ba1\u7b97\u65b9\u6cd5 Numerical method (Spring 2026)/Homework5 \u63d0\u4ea4\u540d\u5355","pageid":2028,"logpage":2028,"revid":13758,"params":{},"type":"create","action":"create","user":"Houzhe","timestamp":"2026-05-21T06:01:02Z","comment":"Created page with \" \u5982\u6709\u9519\u6f0f\u8bf7\u90ae\u4ef6\u8054\u7cfb\u52a9\u6559. <center> {| class=\"wikitable\" |- ! \u5b66\u53f7 !! \u59d3\u540d |- | 211502017 || \u8463\u79d1\u82c7  |- | 221220104 || \u5218\u5b87\u5e73  |- | 231250084 || \u8c22\u94a6\u714c  |- | 231502006 || \u6f58\u8662\u5955  |- | 231502021 || \u674e\u601d\u54f2  |- | 231820107 || \u5f20\u82cf\u7545  |- | 241098018 || \u5434\u7693  |- | 241220023 || \u9648\u5929\u9aa2  |- | 241220026 || \u5f90\u6d69\u7136  |- | 241220028 || \u5468\u65b9\u88d5  |- | 241220043 || \u5f20\u6d9b  |- | 241220058 || \u9648\u661f\u5b87  |- | 241220073 || \u738b\u5b50\u58a8  |- | 241220085 |...\""},{"logid":1527,"ns":6,"title":"File:Computational Method 2026 Assignments 6.pdf","pageid":2025,"logpage":2025,"revid":13755,"params":{"img_sha1":"l1sxpzs2v7u0sfv12o5vjwf5zrbpx5z","img_timestamp":"2026-05-20T13:39:47Z"},"type":"upload","action":"overwrite","user":"Houzhe","timestamp":"2026-05-20T13:39:47Z","comment":""},{"logid":1526,"ns":0,"title":"\u7ec4\u5408\u6570\u5b66 (Fall 2026)/Ramsey theory","pageid":2027,"logpage":2027,"revid":13754,"params":{},"type":"create","action":"create","user":"Etone","timestamp":"2026-05-20T13:33:09Z","comment":"Created page with \"== Ramsey's Theorem == === Ramsey's theorem for graph === {{Theorem|Ramsey's Theorem| :Let <math>k,\\ell</math> be positive integers. Then there exists an integer <math>R(k,\\ell)</math> satisfying: :If <math>n\\ge R(k,\\ell)</math>, for any coloring of edges of <math>K_n</math> with two colors red and blue, there exists a red <math>K_k</math> or a blue <math>K_\\ell</math>. }} {{Proof| We show that <math>R(k,\\ell)</math> is finite by induction on <math>k+\\ell</math>. 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