Chromatic Ramsey Number and Circular Chromatic Ramsey Number. Xuding Zhu. Department of Mathematics Zhejiang Normal University. Among 6 people,. There are 3 know each other, or 3 do not know each other. Know each other. Do not know each other. Among 6 people,.
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Chromatic Ramsey Number and
Circular Chromatic Ramsey Number
Xuding Zhu
Department of Mathematics
Zhejiang Normal University
Among 6 people,
There are 3 know each other,
or 3 do not know each other.
Know each other
Do not know each other
Among 6 people,
There are 3 know each other,
or 3 do not know each other.
Among 6 people,
There are 3 know each other,
or 3 do not know each other.
Among 6 people,
There are 3 know each other,
or 3 do not know each other.
Colour the edges of by red or blue,
there is either a red or a blue
Among 6 people,
There are 3 know each other,
or 3 do not know each other.
For `any’ systems , there exists
a system F such that if `elements’ of F are partitioned
into k parts, then for some i, the ith part contains
as a subsystem.
Theorem [Ramsey] For any graphs G and H, there exists
a graph F such that if the edges of F are coloured by red
and blue, then there is a red copy of G or a blue copy of H
Sufficiently large or complicated
General Ramsey Type Theorem:
A sufficiently largescale (or complicated) systemmust contains
an interesting sub-system.
“Complete disorder is impossible”
There are Ramsey type theorems in many branches
of mathematics such as combinatorics, number theory, geometry,
ergodic theory, topology, combinatorial geometry,
set theory, and measure theory.
Ramsey Theory has a wide range of applications.
Theorem [Ramsey, 1927]
If the k-tuples M are t-colored, then
all the k-tuples of M’ having the same color.
whenever
the
elements
of
some
(suﬃciently
large)
object
are
partitioned
into
a
ﬁnite
number
of
classes
(i.e.,
colored
with
a
ﬁnite
number
of
colors),
there
is
always
at
least
one
(color)
class
which
contains
all
the
elements
of
some
regular
structure.
When
this
is
the
case,
one
additionally
would
like
to
have
quantitative
estimates
of
what
“suﬃciently
large”
means.
In
this
sense,
the
guiding
philosophy
of
Ramsey
theory
can
be
described
by
the
phrase:
“Complete
disorder
is
impossible”
.
Van der Waeden Theorem
For any partition of integers into finitely many parts,
one part contains arithematical progression of arbitrary
large length.
Regularity lemma
Erdos and Turan conjecture (1936)
Szemerédi's theorem (1975)
Every set of integers A with positive density contains
arithematical progressionof arbitrary length.
Harmonic analysis
Timonthy Gowers[2001] gave a proof
using both Fourier analysis and combinatorics.
Ramsey number R(3,k)
Furstenberg
[124]
gave
ergodic
theoretical
and
topological
dynamics
reformulations.
means.
For any 2-colouring of the edges of F with coloursredandblue,
there is a red copy ofGor a blue copy ofH.
The Ramsey number of (G,H) is
1933, George Szekeres, Esther Klein, Paul Erdos
starting with a geometric problem, Szekeres re-discovered
Ramsey theorem, and proved
Szekere [1933]
Erdos [1946]
Erdos [1961]
Graver-Yackel [1968]
Ajtai-Komlos-Szemeredi
[1980]
Kim [1995]
Many sophisticated probabilistic tools are developed
George Szekere and Esther Klein married
lived together for 70 year,
died on the same day 2005.8.28, within one hour.
A sufficiently largescale (or complicated) systemmust contains
an interesting sub-system.
How to measure a system?
What is large scale?
What is complicated?
How to measure a graph?
Chromatic number
Circular chromatic number
G=(V,E): a graph
0
an integer
1
1
An k-colouring of G is
2
0
such that
A 3-colouring of
The chromatic number of G is
G=(V,E): a graph
0
a real number
1
an integer
1.5
A (circular)
k-colouring of G is
r-colouring of G is
An
2
0.5
A 2.5-coloring
such that
The circular chromatic number of G is
{ r: G has a circular r-colouring }
min
inf
f is k-colouring of G
f is a circular k-colouring of G
Therefore for any graph G,
0=r
0
r
1
4
2
3
|f(x)-f(y)|_r ≥ 1
x~y
p
p’
The distance between p, p’ in the circle is
f is a circular r-colouring if
and
Basic relation between
Circular chromatic number of a graph is a refinement
of its chromatic number.
Graph coloring is a model for resource distribution
Circular graph coloring is a model for resource distribution
of periodic nature.
Introduced by Burr-Erdos-Lovasz in 1976
If F has chromatic number , then there is a
2 edge colouring of F in which each monochromatic
subgraph has chromatic number n-1.
for any n-chromatic G.
If F has chromatic number , then there is a
2 edge colouring of F in which each monochromatic
subgraph has chromatic number n-1.
for any n-chromatic G.
Could be much larger
There are some upper bounds on
The conjecture is true for n=3,4 (Burr-Erdos-Lovasz, 1976)
The conjecture is true for n=5 (Zhu, 1992)
Attempts by Tardif, West, etc. on non-diagonal cases
of chromatic Ramsey numbers of graphs.
No more other case of the conjecture were verified, until 2011
The conjecture is true (Zhu, 2011)
For any 2 edge-colouring of Kn, there is a monochromatic
graph which is a homomorphic image of G.
Graph homomorphism = edge preserving map
H
G
To prove Burr-Erdos-Lovasz conjecture for n, we need to construct
an n-chromatic graph G, so that any 2 edge colouring of
has a monochromatic subgraph which is a homomorphic image of G.
The construction of G is easy:
Take all 2 edge colourings of
For each 2 edge colouring ci of , one of the monochromatic
subgraph, say Gi, , has chromatic number at least n.
To prove this conjecture for n, we need to construct an
n-chromatic graph G, so that any 2 edge colouring of
has a monochromatic subgraph which is a homomorphic image of G.
The construction of G is easy:
Take all 2 edge colourings of
For each 2 edge colouring of , one of the monochromatic
subgraph, say Gi, , has chromatic number at least n.
H
G
GxH
Projections are homomorphisms
To prove this conjecture for n, we need to construct an
n-chromatic graph G, so that any 2 edge colouring of
has a monochromatic subgraph which is a homomorphic image of G.
?
The construction of G is easy:
Take all 2 edge colourings of
For each 2 edge colouring ci of , one of the monochromatic
subgraph, say Gi, , has chromatic number at least n.
H
G
To prove this conjecture for n, we need to construct an
n-chromatic graph G, so that any 2 edge colouring of
has a monochromatic subgraph which is a homomorphic image of G.
?
If Hedetniemi’s conjecture is true, then
Burr-Erdos-Lovasz conjecture is true.
A k-colouring of G partition V(G) into k independent sets.
integer linear programming
A k-colouring of G partition V(G) into k independent sets.
linear programming
Fractional Hedetniemi’s conjecture
Observation: If fractional Hedetniemi’s conjecture is true,
then
Burr-Erdos-Lovasz conjecture is true.
To prove this conjecture for n, we need to construct an
n-chromatic graph G, so that any 2 edge colouring of
has a monochromatic subgraph which is a homomorphic image of G.
If Hedetniemi’s conjecture is true, then
Burr-Erdos-Lovasz conjecture is true.
To prove this conjecture for n, we need to construct an
n-chromatic graph G, so that any 2 edge colouring of
has a monochromatic subgraph which is a homomorphic image of G.
The construction of G is easy:
Take all 2 edge colourings of
For each 2 edge colouring ci of , one of the monochromatic
subgraph, say Gi, , has chromatic number at least n.
fractional chromatic number > n-1
Theorem [Huajun Zhang, 2011]
If both G and H are vertex transitive, then
Fractional Hedetniemi’s conjecture
Theorem [Z, 2011]
A k-colouring of G partition V(G) into k independent sets.
dual problem
linear programming
The fractional chromatic number of G is obtained
by solving a linear programming problem
The fractional clique number of G is obtained
by solving its dual problem
Fractional Hedetniemi’s conjecture is true
Theorem [Z, 2010]
Easy!
Difficult!
Easy
Easy
Difficult
Difficult!
Easy!
What is the relation between and ?
is a refinement of
and
Basic relation between
is an approximation of
The reciprocal of is studied by computer
scientists as efficiency of a certain
scheduling method, in 1986.
Circular colouring is a good model for periodical
scheduling problems
There are many periodical scheduling problems in
computer sciences.
Theorem [Zhu, 2011]
No conjecture yet!
Using fractional version of Hedetniemi’s conjecture,
Jao-Tardif-West-Zhu proved in 2014
min
?
min
No !
[ Jao-Tardif-West-Zhu, 2014]
Some other results by Jao-Tardif-West-Zhu, 2014
謝謝