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Ramsey Properties

Proven via. A Hypergraph Regularity Lemma. Ramsey Properties. of Random Graphs;. A Sharp Threshold. Ehud Friedgut, Vojtech Rödl, Andrzej Rucinski, Prasad Tetali. Special thanks to the Tetali family for c o s t u m e d e s i gn. Chapter I. What makes a graph Ramsey?. Why?!.

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Ramsey Properties

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  1. Proven via A Hypergraph Regularity Lemma. RamseyProperties of Random Graphs; A Sharp Threshold Ehud Friedgut, Vojtech Rödl, Andrzej Rucinski, Prasad Tetali

  2. Special thanks to the Tetali family for costumedesign.

  3. Chapter I What makes a graph Ramsey?

  4. Why?! We will say a graph is a Ramsey graph if every bi-coloring of its edges contains a monochromatic triangle. e.g.

  5. Is there a sharp threshold?

  6. Theorem:Yes, there does.

  7. Why is the critical edge probability? The expected number of triangles per edge is

  8. Chapter II Regularity and Cores

  9. A multi-partite graph on vertex sets is -regular if all but of the pairs are -regular Easy if k is very small or very large…

  10. Szemerédi’s Regularity Lemma: Weighted variations? Sparse graphs? Hypergraphs?

  11. A hitting set of a graph G is a set of vertices that intersects every edge. In a dense graph on vertices there may be hitting sets. 2. Thenumberof cores is . 3. Every core is of size linear in . We would like to capture all hitting sets by a family of cores so that: • Every hitting set contains a core.

  12. U V If G is -regular bipartite take all sets U’ or V’ such that or • Every hitting set contains a core. 3. Every core is of size linear in . 2. The number of cores is . If G is a complete bipartite graph on vertex sets U, V take the cores to be U and V.

  13. A core will be any set obtained by taking a hitting set in the super-graph and taking at least of the vertices in all the super-vertices involved. In a general graph – fix a Szemerédi partition. Draw the super-graph of regular pairs.

  14. Chapter III Proving a threshold is sharp

  15. “Theorem”: Sharp threshold Global property Coarse threshold Local property e.g. connectivity has a sharp threshold - whereas containing a triangle has a coarse threshold.

  16. Any such would be sensitive to small global enhancement … …which means exactly that Ramsiness has asharp threshold! If Ramsiness had a coarse threshold it would be local – a typical non-Ramsey would be sensitive to local perturbations…

  17. Let be typical in . Assume is non-Ramsey. • Assume there exists a small magical graph , say , such that • Show that this implies

  18. is not seen in ! What about ?

  19. Many copies of will pose restrictions if they appear – e.g. a problematic copy: But in every proper coloring of one of the following will happen: We can color

  20. B BB R R Every restriction consists of five elements such that every proper coloring must agree with on at least one of them. Using probabilistic techniques we can arrange a large subset of these restrictions as follows:

  21. This defines a hypergraph with (hypergraph)edges of size 5. : B BB R R Given a proper coloring of , and an edge of then there exists an element in for which agrees with . For every proper coloring , the set of (graph)edges of on which it agrees with is a hitting set of .

  22. Every large partial coloring survives the addition of a random copy of with probability . There are approximately colorings. Union bound: Depends on the value of ! There may be too many colorings. How does one show sensitivity to global enhancement?

  23. Last chapter: Putting it all together

  24. We have a hypergraph of restrictions such that every proper coloring defines a hitting set of . But, there are too many colorings. We would like to capture them by a family of cores such that : 2. The number of cores is . 3. Every core is of size . 1. Every hitting set contains a core. We then can improve the union bound by clumping:

  25. Survival probability of each. Colorings (hitting sets) All these colorings share a core. Cores There are many colorings :

  26. Partition the vertices of (auxiliary partition) A Frankl – Rödl partition 2. Partition every one of the bipartite graphs formed into (non-induced) subgraphs.

  27. Choosing five of these bipartite graphs and a subgraph of each gives a polyad, a set of 5 subsets (anologous to a pair of sets in a Szemerédi partition.)

  28. The density of a polyad The number of copies of belonging to = (The total number of copies of) A regular Polyad – every sufficiently “large” subgraph has density close to that of the polyad.

  29. “Theorem”: If is a typical graph in and is the corresponding restriction hypergraph then there exists a Frankl-Rödl partition of such that “most” of the polyads formed are -regular. This enables us to define cores, capture all colorings efficiently and finish the proof.

  30. Believe me, you don’t want to know. So, what is the definition of a core?

  31. Epilogue

  32. And In conclusion I would like to say:

  33. Thank you for your attention!! • Ramsiness has a sharp threshold because it is a global property. Union bounds can be improved by clumping • Clumping can be done if the underlying structure has an inherent regularity. • Frankl –Rödl type partitions can extract regularity from various hypergraphs.

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