Ramsey Theory
Definition    • Ramsey theory ask: "how many elements of      some structure must there be to guarantee that      a partic...
Basics    • Graphs          – Edges          – Vertices          – Edge colorings    • Complete graphs    • PigeonholeMar ...
Mar 14, 2012   4
Pigeonhole• Pigeonhole is a simple theory       – m objects divide into n classes       – at least [m/n] objects appears• ...
Example: Erdὀs Szekeres Theorem• Given a sequence of mn+1 distinct real  numbers, it either contains a monotone  increasin...
Proof:• By contradiction• S := {a1,…,amn+1 } be distinct mn+1 numbers.• Each number ak has a tuple (i , j).  i – length of...
Cont..• 2 elements S and T such that tuple is same.  Let it be (x, y).• Assume S < T.  Then if T has a increasing sequence...
Ramsey Number• Ramsey numbers part of mathematical field of graph  theory• km is defined as a graph containing m nodes and...
Ramsey Number  • The Ramsey number R(m , n)gives the solution    to the party problem, which asks the minimum    number of...
Ramsey Number               Example: R(3,3) = 6                Possible to create a        Not possible to create a       ...
Ramsay Theorem• In the language of graph theory, the Ramsey  number is the minimum number of vertices  v=R(m,n),such that ...
Existence Proof of R(r,s)• for the 2-colour case, by induction on r + s• Existance proof by proving a explicit bound• base...
Existence Proof of R(r,s)• Consider a complete graph on R(r − 1, s) + R(r, s − 1) vertices• Pick a vertex v from the graph...
February 24, 2007   15
Ramsey Numbers• R(4,3)<=R(3,3) + R(4,2)        <= 6 + 4 = 10• R(4,3) = 9Mar 14, 2012
Graph Ramsey Number• Given simple graphs G1,…,Gk,the graph  Ramsey number R(G1,…,Gk) is the smallest  integer n such that ...
Ramsey number    • R(M1,M2,…,Mk )    • When k>2, little is known.           – R(3,3,3)=14    • Ramsey number tell us that ...
Other Ramsey Theory•    Graph Ramsey Number•    Ramsey Polygon Number•    Ramsey of Bipartite graph•    ……Mar 14, 2012    ...
Ramsey Theory ApplicationsMar 14, 2012                 20
Final Thoughts• Results in Ramsey theory typically have two  primary characteristics:       – non-constructive: exist but ...
Final Thoughts…    The reason behind such Ramsey-type results is that:    “The largest partition class always contains the...
REFERENCES• Ramsey Theory and Related Topics (Fall 2004, 2.5 cu) J.  Karhumaki• Introduction to Graph Theory by Douglas B....
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Tapas seminar

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Tapas seminar

  1. 1. Ramsey Theory
  2. 2. Definition • Ramsey theory ask: "how many elements of some structure must there be to guarantee that a particular property will hold?“Mar 14, 2012 2
  3. 3. Basics • Graphs – Edges – Vertices – Edge colorings • Complete graphs • PigeonholeMar 14, 2012 3
  4. 4. Mar 14, 2012 4
  5. 5. Pigeonhole• Pigeonhole is a simple theory – m objects divide into n classes – at least [m/n] objects appears• Pigeonhole and Ramsey are closely linked – Some Ramsey can be proved by Pigeonhole – They both satisfy “how many elements can guarantee a property”• Sometimes deceivingMar 14, 2012 5
  6. 6. Example: Erdὀs Szekeres Theorem• Given a sequence of mn+1 distinct real numbers, it either contains a monotone increasing subsequence of length m + 1 or a monotone decreasing subsequence of length n + 1.Mar 14, 2012 6
  7. 7. Proof:• By contradiction• S := {a1,…,amn+1 } be distinct mn+1 numbers.• Each number ak has a tuple (i , j). i – length of longest increasing sequence starting from ak j – length of longest decreasing sequence starting from ak .• Assumption– 1≤ i ≤ m 1≤ j ≤ nMar 14, 2012 7
  8. 8. Cont..• 2 elements S and T such that tuple is same. Let it be (x, y).• Assume S < T. Then if T has a increasing sequence of length x, then S will have a increasing sequence of x+1.Mar 14, 2012 8
  9. 9. Ramsey Number• Ramsey numbers part of mathematical field of graph theory• km is defined as a graph containing m nodes and all possible line between the nodes• Ramsey functions notated as R(r, b)=n – R is Ramsey function – r, b are independent variables – n is result of Ramsey function; called Ramsey number• Ramsey function gives smallest graph size that when colored in any pattern of only two colors, will always contain a monochromatic kr or a monochromatic kb .Mar 14, 2012 9
  10. 10. Ramsey Number • The Ramsey number R(m , n)gives the solution to the party problem, which asks the minimum number of guests R(m , n) that must be invited so that at least m are mutual friends or at least n are mutual strangers. • R(X,1)=R(1,X)=1 • R(2,X)=X • R(3,3)=??Mar 14, 2012 10
  11. 11. Ramsey Number Example: R(3,3) = 6 Possible to create a Not possible to create a mapping without a k3 in a coloring without a k3 in a 6- 5-node graph (a k5) node graph (a k6) and this R(3,3) = 6Mar 14, 2012 11
  12. 12. Ramsay Theorem• In the language of graph theory, the Ramsey number is the minimum number of vertices v=R(m,n),such that all undirected simple graphs of order v contain a clique of order m or an independent set of order n .• Ramsey theorem states that such a number exists for all m and n.Mar 14, 2012 12
  13. 13. Existence Proof of R(r,s)• for the 2-colour case, by induction on r + s• Existance proof by proving a explicit bound• base: for all n, R(n, 1) = R(1, n) = 1• By the inductive hypothesis R(r − 1, s) and R(r, s − 1) exist.• Claim: R(r, s) ≤ R(r − 1, s) + R(r, s − 1)Mar 14, 2012 13
  14. 14. Existence Proof of R(r,s)• Consider a complete graph on R(r − 1, s) + R(r, s − 1) vertices• Pick a vertex v from the graph, and partition the remaining vertices into two sets M and N, such that for every vertex w, w is in M if (v, w) is blue, and w is in N if (v, w) is red.• Because the graph has R(r - 1, s) + R(r, s - 1) = |M| + |N| + 1 vertices, it follows that either |M| ≥ R(r − 1, s) or |N| ≥ R(r, s − 1)• In the former case, if M has a red Ks then so does the original graph and we are finished.• Otherwise M has a blue Kr−1 and so M U {v} has blue Kr by definition of M. The latter case is analogous.• R(4,3)=??Mar 14, 2012 14
  15. 15. February 24, 2007 15
  16. 16. Ramsey Numbers• R(4,3)<=R(3,3) + R(4,2) <= 6 + 4 = 10• R(4,3) = 9Mar 14, 2012
  17. 17. Graph Ramsey Number• Given simple graphs G1,…,Gk,the graph Ramsey number R(G1,…,Gk) is the smallest integer n such that every k-coloring of E(Kn) contains a copy of Gi in color i for some i.Mar 14, 2012 17
  18. 18. Ramsey number • R(M1,M2,…,Mk ) • When k>2, little is known. – R(3,3,3)=14 • Ramsey number tell us that R(m1,m2,…,mk) always exist!Mar 14, 2012 18
  19. 19. Other Ramsey Theory• Graph Ramsey Number• Ramsey Polygon Number• Ramsey of Bipartite graph• ……Mar 14, 2012 19
  20. 20. Ramsey Theory ApplicationsMar 14, 2012 20
  21. 21. Final Thoughts• Results in Ramsey theory typically have two primary characteristics: – non-constructive: exist but non-consturctive • This is same for pigeonhole – Grow exponetially: results requires these objects to be enormously large. • That’s why we still know small ramsey number • Computer is useless here!Mar 14, 2012 21
  22. 22. Final Thoughts… The reason behind such Ramsey-type results is that: “The largest partition class always contains the desired substructure”.Mar 14, 2012 22
  23. 23. REFERENCES• Ramsey Theory and Related Topics (Fall 2004, 2.5 cu) J. Karhumaki• Introduction to Graph Theory by Douglas B.West , 2-ed• Applications of Discrete Mathematics by John G. Michaels ,Kenneth H.Rosen• http://en.wikipedia.org/wiki/Ramseys_theorem• Noga Alon and Michael Krivelevich [The Princeton Companion to Mathematics]• Ramsey Theory Applications:Vera RostaMar 14, 2012 23
  24. 24. Questions
  25. 25. Thank you

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