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Upper bounds for asymmetric Ramsey properties of random graphs. Reto Spöhel, ETH Zürich Joint work with Yoshiharu Kohayakawa, Universidade de S ã o Paulo Mathias Schacht, Universität Hamburg. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A.

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upper bounds for asymmetric ramsey properties of random graphs

Upper bounds for asymmetric Ramsey properties of random graphs

Reto Spöhel, ETH Zürich

Joint work with

Yoshiharu Kohayakawa, Universidade de São PauloMathias Schacht, Universität Hamburg

TexPoint fonts used in EMF.

Read the TexPoint manual before you delete this box.: AAAAA

ramsey theory
Ramsey Theory
  • Folklore
    • Among every party of at least six people, there are at least three, either all or none of whom know each other
    • Equivalently:Every edge-coloring of the complete graph on six vertices with two colors contains a monochromatic triangle.
  • Question:
    • How many people must attend the party so that the assertion holds for ` > 3people?
    • Are these numbers finite?
ramsey theory1
Ramsey Theory

Ramsey (1930)

  • Extensions:
    • Color graphs other than cliques (e.g., random graphs).
    • Avoid some fixed graph Fother thanK`.
    • Avoid graph F1 in blue and F2 in red (asymmetric case).
    • Allow more colors.
random graphs
Random Graphs
  • Binomial model Gn,p
    • n vertices
    • include each edge with probability p, independently of all other edges
  • We study the limiting probabilitythat the random graph Gn,p satisfies a given property P, where p=p(n).
  • It turns out that many properties have threshold functionsp0(n) such that
ramsey properties
Ramsey properties

R(F, k)

  • Problem: For any fixed graph F, integer k, and edge probability p=p(n), determine
  • Observation:
    • The family of graphs satisfying R(F, k) is increasing.
    •  The property R(F, k) has a threshold (Bollobás, Thomason, 1987).

By R(F, k) we denote the family of all graphs that contain a monochromatic copy of F in every edge coloring with k colors.

threshold for ramsey properties

Łuczak, Ruciński, Voigt (1992)/Rödl, Ruciński (1993, 1995)

Threshold for Ramsey properties
  • Intuition:
    • above the threshold, there are more copies of Fin Gn,p than edges.
    • This forces the copies of F to overlap substantially and makes coloring difficult.
  • Order of magnitude of threshold does not depend on k (!)
asymmetric ramsey properties

R(G, H)

ByR(G, H) we denote the family of all graphs that contain either a red copy of Gor a blue copy of Hin every edge coloringwith red and blue.

Asymmetric Ramsey properties
  • What happens is if we want to avoid differentgraphs Fi in different colors i, 1·i·k?
    • We focus on the case with two colors.
threshold for asymmetric ramsey properties
Threshold for asymmetric Ramsey properties

Conjecture: Kohayakawa, Kreuter (1997)

Kohayakawa, Kreuter (1997)

The conjecture is true if G and H are cycles.

Marciniszyn, Skokan, S., Steger (RANDOM’06)

The 0-statement is true if G and H are cliques.

threshold for asymmetric ramsey properties1
Threshold for asymmetric Ramsey properties

Conjecture: Kohayakawa, Kreuter (1997)

(e.g.) Marciniszyn, Skokan, S., Steger (RANDOM’06)

The 1-statement is true if H satisfies some balancedness

condition and the KŁR-Conjecture holds for G.

  • The KŁR-Conjecture is known to be true for trees, cycles, and cliques of size at most 5.
threshold for asymmetric ramsey properties2
Threshold for asymmetric Ramsey properties

Conjecture: Kohayakawa, Kreuter (1997)

Kohayakawa, Schacht, S. (2009+)

The 1-statement is true if H satisfies some balancedness

condition and the KŁR-Conjecture holds for G.

  • …. in particular for the case where G and H are cliques of arbitrary size.
overview
Overview
  • Ourproofisalongsimilarlines as the original proofforthesymmetriccaseby Rödl and Ruciński.
    • Need toshow: withhighprobability, in eachcoloringofGn,pthereiseither a bluecopyofHor a redcopyofG.
  • Weshowbyinduction on e(G): withveryhighprobability, in eachcoloringof[…] (e.g. Gn,p) thereiseither a bluecopyof H ormany […] redcopiesofG.
    • denotethiseventbyR
  • Wesolveonekeyissuein a fundamentally different waythan in the original proof.
    • also yields a simpler proofforthesymmetriccase
small and very small
„Small“ and „very small“
  • Terminology:
    • „small“: something like
    • „very small“: something like
the issue
The issue
  • Goal: show that Pr[:R] is very small (needed for induction).
  • At some point in the proof, we need that there are not too many copies of some graph D.

many (byinduction)

not toomany !?

the issue1
The issue
  • Goal: show that Pr[:R] is very small (needed for induction).
  • At some point in the proof, we need that there are not too many copies of some graph D.
    • denote this event by D
  • Assuming that Dholds, we can show that the probability for :R is indeed „very small“.
    • i.e. we can show
  • Issue: Pr[:D] is „small“, but not „very small“!
    • We only get
the deletion method
The deletion method
  • Rödl and Ruciński (1995): Maybedeleting a fewedgessufficestoensurethattherearenot toomanycopiesofD in theremaining graph.
    • denotethiseventbyD*
  • Deletion Lemma: Pr[:D*] isindeed „verysmall“
  • Robustness Lemma:Deleting a fewedgesdoes not mess uptherestoftheproof.
    • i.e. wecan still show
  • Weget
alternative solution
Alternative solution
  • Use the FKG inequality!
  • A graph property is called increasing if it cannot be destroyed by adding edges
    • containing a C4 is an increasing property
    • containing an inducedC4 is not an increasing property.
alternative solution1
Alternative solution
  • Use the FKG inequality!
  • A graph property is called increasing if it cannot be destroyed by adding edges
  • not true for arbitrary distributions!
    • in particular not for Gn,m !

Fortuin, Kasteleyn, Ginibre (1971)

alternative solution2
Alternative solution
  • Similarly:
  • A graph property is called decreasing if it cannot be destroyed by removing edges
      • A decreasing ,:Aincreasing.

Fortuin, Kasteleyn, Ginibre (1971)

alternative solution3
Alternative solution
  • Goal: show that Pr[:R] is „very small“
  • Can show: Pr[:R Æ D] is „very small“
  • Issue: Pr[:D] is „small“, but not „very small“!
  • Observation: Both :R and Dare decreasing events!
summary

Marciniszyn, Skokan, S., Steger (2006) / Kohayakawa, Schacht, S. (2009)

Summary
  • We proved an upper bound on the threshold for a large class of asymmetric Ramsey properties.
  • Together with earlier results, we obtain in particular
  • In our proof we replaced the deletion method by an „FKG shortcut“. This also yields a simpler proof for the original symmetric case.
  • Our proof seems to extend to more than two colors, but this needs considerable extra work.
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