Difference between revisions of "Ramsey Number"
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PhoebeJiang (talk  contribs) 

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==Definition==  ==Definition==  
−  Ramsey number is the solution to the party problems, which ask the minimum number of guests that must be invited so that at least m will know each other or at least n will not know each other.  +  Ramsey number <math>R(m, n)</math> is the solution to the party problems, which ask the minimum number of guests that must be invited so that at least <math>m</math> will know each other or at least <math>n</math> will not know each other. 
−  ==  +  ==A Summary of Known Ramsey Numbers== 
+  
+  {{{!}} class="wikitable"  
+  ! ''r'',''s''  
+  ! 1  
+  ! 2  
+  ! 3  
+  ! 4  
+  ! 5  
+  ! 6  
+  ! 7  
+  ! 8  
+  ! 9  
+  ! 10  
+  {{!}}  
+  ! 1  
+  {{!}} 1  
+  {{!}} 1  
+  {{!}} 1  
+  {{!}} 1  
+  {{!}} 1  
+  {{!}} 1  
+  {{!}} 1  
+  {{!}} 1  
+  {{!}} 1  
+  {{!}} 1  
+  {{!}}  
+  ! 2  
+   1  
+   2  
+   3  
+   4  
+   5  
+   6  
+   7  
+   8  
+   9  
+   10  
+    
+  ! 3  
+   1  
+   3  
+   6  
+   9  
+   14  
+   18  
+   23  
+   28  
+   36  
+   40–43  
+    
+  ! 4  
+   1  
+   4  
+   9  
+   18  
+   25  
+   35–41  
+   49–61  
+   56–84  
+   73–115  
+   92–149  
+    
+  ! 5  
+   1  
+   5  
+   14  
+   25  
+   43–49  
+   58–87  
+   80–143  
+   101–216  
+   125–316  
+   143–442  
+    
+  ! 6  
+   1  
+   6  
+   18  
+   35–41  
+   58–87  
+   102–165  
+   113–298  
+   127–495  
+   169–780  
+   179–1171  
+    
+  ! 7  
+   1  
+   7  
+   23  
+   49–61  
+   80–143  
+   113–298  
+   205–540  
+   216–1031  
+   233–1713  
+   289–2826  
+    
+  ! 8  
+   1  
+   8  
+   28  
+   56–84  
+   101–216  
+   127–495  
+   216–1031  
+   282–1870  
+   317–3583  
+   3176090  
+    
+  ! 9  
+   1  
+   9  
+   36  
+   73–115  
+   125–316  
+   169–780  
+   233–1713  
+   317–3583  
+   565–6588  
+   580–12677  
+    
+  ! 10  
+   1  
+   10  
+   40–43  
+   92–149  
+   143–442  
+   179–1171  
+   289–2826  
+   3176090  
+   580–12677  
+   798–23556  
+  }  
==Examples==  ==Examples== 
Revision as of 15:55, 21 July 2011
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Pigeonhole Principle 
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Definition
Ramsey number is the solution to the party problems, which ask the minimum number of guests that must be invited so that at least will know each other or at least will not know each other.
A Summary of Known Ramsey Numbers
r,s  1  2  3  4  5  6  7  8  9  10 

1  1  1  1  1  1  1  1  1  1  1 
2  1  2  3  4  5  6  7  8  9  10 
3  1  3  6  9  14  18  23  28  36  40–43 
4  1  4  9  18  25  35–41  49–61  56–84  73–115  92–149 
5  1  5  14  25  43–49  58–87  80–143  101–216  125–316  143–442 
6  1  6  18  35–41  58–87  102–165  113–298  127–495  169–780  179–1171 
7  1  7  23  49–61  80–143  113–298  205–540  216–1031  233–1713  289–2826 
8  1  8  28  56–84  101–216  127–495  216–1031  282–1870  317–3583  3176090 
9  1  9  36  73–115  125–316  169–780  233–1713  317–3583  565–6588  580–12677 
10  1  10  40–43  92–149  143–442  179–1171  289–2826  3176090  580–12677  798–23556 