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Introduction Perturb I Perturb II Beating random Conclusion
Finding patterns avoiding manymonochromatic constellations
Steve Butler1
(joint work with Kevin Costello and Ron Graham)
1Department of MathematicsUniversity of California Los Angeleswww.math.ucla.edu/~butler
2009 Fall Western Section Meeting7 November 2009
Introduction Perturb I Perturb II Beating random Conclusion
Can we color the integers [n] = {1,2, . . . ,n} with colors REDand BLUE so that there is no three term arithmetic progressionall red or all blue?
Theorem (van der Waerden)
For fixed k and r , any coloring of [n] with n ≥W (r , k) using rcolors contains an arithmetic progression of length k .
Corollary
For fixed k and r , there is a γ > 0 so that any coloring of [n]using r colors contains at least
(γ + o(1)
)n2 different
monochromatic k -term arithmetic progressions.
Introduction Perturb I Perturb II Beating random Conclusion
How sparsely can we pack three term arithmetic progressionsinto a two coloring of [n]?
What is the minimal γ so that a coloring of [n] with two colorscontains (
γ + o(1))n2
three term arithmetic progressions?
First guess: Color randomly. γ = 116
(There are ≈ n2/4 three term arithmetic progressions, in arandom coloring a fixed progression has probability of 1/4 ofbeing monochromatic so we expect ≈ n2/16.)
Introduction Perturb I Perturb II Beating random Conclusion
A pattern which beats random!
This consists of 12 blocks with relative sizes:28-6-28-37-59-116-116-59-37-28-6-28
For this coloring we have
γ =1172192
≈ 0.05337591 . . . <1
16= 0.0625.
(Found independently by B-C-G andParrilo-Robertson-Saracino.)
How did we find this pattern? How did we calculate γ?Why do we think this might be the best pattern?
Introduction Perturb I Perturb II Beating random Conclusion
Constellations
A constellation pattern is a collection of rationals qi ∈ [0,1]which includes 0 and 1. A constellation is a scaled translatedrealization in [n].
{0, 12,1} ↔ three term AP
{0, 13,23,1} ↔ four term AP
{0, 14,12,34,1} ↔ five term AP
{0, 13,1} ↔ solutions to x + 2y = 3z
{0, 25,1} ↔ solutions to 2x + 3y = 5z
{0, aa + b
,1} ↔ solutions to ax + by = (a + b)z
Introduction Perturb I Perturb II Beating random Conclusion
ProblemGiven a constellation pattern, find a small (ideally, smallest) γso that there is a two coloring of [n] with(
γ + o(1))n2
monochromatic constellations, and also give the coloring.
As an example we will use the constellation {0, 13 ,1}.
Introduction Perturb I Perturb II Beating random Conclusion
ObservationIf we are in a coloring of [n] which minimizes the number ofmonochromatic constellations, then switching the color of anysingle entry will not cause the number of monochromaticconstellations to decrease.
Perturbate to find (local) minimal colorings
Given a two coloring of [n].Test to see if changing the color of a given numberdecreases the number of monochromatic constellations. Ifit does, change it.Go to the next number.Stop when changing the coloring of any given number doesnot decrease the number of monochromatic solutions.
Introduction Perturb I Perturb II Beating random Conclusion
There are two major decisions in applying this procedure:What coloring do we begin with?How do we go to the next element (i.e., sweep back andforth, random, etc.)?
When we stop we will be at a local minimal, but this might be farfrom the global minimal. There are several ways to deal withthis.
Run the experiment many times with varying startingpatterns and rules for moving to the next element and takethe best one that is produced.Run the experiment, when it stops hit it with a smallperturbation and then keep running it. Repeat severaltimes.
Introduction Perturb I Perturb II Beating random Conclusion
Testing the perturbation
We can test this procedure on Schur triples (x + y = z) forwhich it is known that their are at least 1
22n2 + O(n)monochromatic solutions and that the best coloring thatachieves this is a block coloring with three blocks with relativesizes 4-6-1. (Robertson-Zeilberger and Schoen)
Introduction Perturb I Perturb II Beating random Conclusion
Starting in blocks
Now we look at runs for the constellation {0, 13 ,1}.
Single block Ten equal blocks
Introduction Perturb I Perturb II Beating random Conclusion
Random starts
Some more runs for the constellation {0, 13 ,1}.
Random Random
Introduction Perturb I Perturb II Beating random Conclusion
Looking at these runs, we see that they all seem to go to apattern with 18 blocks.
Doing a run on [25000] we get block sizes
1101-193-577-583-989-1434-1115-2833-3680-3681-2830-1113-1434-988-582-575-194-1098
These block sizes are approximations for the relative blocksizes in the locally optimal coloring. The next step is to find thecorrect relative block sizes, and the corresponding γ.
Introduction Perturb I Perturb II Beating random Conclusion
Calculating γ given a block pattern
Given a constellation pattern, qi , let D be the smallest commondenominator. Let f : [0,1]→ {±1} by scaling the block patternto the interval [0,1] and sending blue to 1 and red to −1. Then
γ =
α
2Dn2 + O(n) if constellation is symmetric,
α
Dn2 + O(n) if constellation is not symmetric.
where
α =
∫ 1
0
∫ 1
0
(∏i
1 + f(qix + (1− qi)y
)2
+∏
i
1− f(qix + (1− qi)y
)2
)dy dx .
Introduction Perturb I Perturb II Beating random Conclusion
Looking more closely at α
The function that we are integrating over to calculate α,
g(x , y) =∏
i
1 + f(qix + (1− qi)y
)2
+∏
i
1− f(qix + (1− qi)y
)2
is a 0-1 indicator function which indicates monochromaticconstellations inside of [0,1].
The function g(x , y) can only change values when we cross aline of the form
qix + (1− qi)y = βj ,
where βj is where two blocks meet.The nonzero regions of g(x , y) are convex polygons boundedby these lines.
Introduction Perturb I Perturb II Beating random Conclusion
An example for {0, 12 ,1}
A plot of the function g(x , y) is shown below (red and blueindicate where red and blue three term APs are located).
Introduction Perturb I Perturb II Beating random Conclusion
Using the approximate block structure for {0, 13 ,1} we get:
Calculating we have γ =56816777
750000000= 0.07575570 . . ..
Introduction Perturb I Perturb II Beating random Conclusion
ObservationIf our block sizes are locally minimal with respect to calculatingγ, then a small perturbation in one of the βj terms cannotdecrease γ.
(amount of change in redunder ε perturbation of βj
)+
(amount of change in blueunder ε perturbation of βj
)=0
Introduction Perturb I Perturb II Beating random Conclusion
{∆x
qj′x+ (1− qj′)y = βj
qj′x+ (1− qj′)y = βj + ε
qi′x+ (1− qi′)y = βi
qk′x+ (1− qk′)y = βk
∆Area ≈ ∆x1− qj ′
ε
≈(
1qj ′ − qi ′
βi+1
1− qj ′
(1− qk ′
qj ′ − qk ′+
1− qi ′
qi ′ − qj ′
)βj+
1qk ′ − qj ′
βk
)ε.
Introduction Perturb I Perturb II Beating random Conclusion
For a constellation pattern with k points this sets up k linearequations in k unknowns that we can now solve. This gives ourlocally optimal block structure.
For {0, 13 ,1} solving this system gives that the locally optimal
block structure has relative block sizes of
1552213-272415-813251-822338-1394548-2025068-1572841-3995910-5196075-5196075-3995910-1572841-
2025068-1394548-822338-813251-272415-1552213.
Calculating for this pattern we have
γ =16040191
211735908= 0.075755648 . . . .
(Note that randomly we would expect 112 = 0.0833333 . . ..)
Introduction Perturb I Perturb II Beating random Conclusion
Outcome of many experiments
We can now repeat this same procedure for differentconstellation patterns. Below is a chart recording the outcomeof three point constellations {0,q,1} where q is given in thex-axis and on the y -axis we record how well the pattern thatwas discovered did when compared to random coloring.
0.85
0.9
0.95
110
210
310
410
510
610
710
810
910
Introduction Perturb I Perturb II Beating random Conclusion
Random is never best
Given 2a < b, let 0 < ε < 1 + ab −
ab
⌈ba
⌉. Then for the
constellation pattern {0, ab ,1} the coloring found by scaling the
block pattern(1− ε)-(1 + ε)- 1-1- . . . -1︸ ︷︷ ︸
2b − 2 terms
has
γ =
1
4b+
(2a− adb/ae)8ab2(b − a)
ε+ O(ε2) if db/ae odd,
14b
+(a− 2b + adb/ae)
8ab2(b − a)ε+ O(ε2) if db/ae even.
In particular, it beats random!
Introduction Perturb I Perturb II Beating random Conclusion
0.85
0.9
0.95
110
210
310
410
510
610
710
810
910
This appears to be continuous. (Expected since if we fixour block pattern and perturb q by a small amount then wewill only slightly perturb our percentage of random by asmall amount.)The point at q = 2
5 (and by symmetry q = 35 ) seems to be
out of place!
Introduction Perturb I Perturb II Beating random Conclusion
Local perturbation run for {0, 25 ,1}
Introduction Perturb I Perturb II Beating random Conclusion
Alternating blocks for {0, 25 ,1}
Looking at the outcome of these runs we see
↓· · ·RBRBRBRBRRBRBRBRBRBR · · · .
If we look at every other term then we see blocks, the extra Rhas the effect of changing the parity of the location of the Rsand Bs between two blocks. We call these alternating blocks.
Introduction Perturb I Perturb II Beating random Conclusion
Blowing up an alternating block pattern gives us
γ =1
20+
18
∫ 1
0
∫ (2+3y)/5
3y/5f (x)f (y) dx dy .
This is the same as minimizing the amount of red inside aparallelogram as shown below.
Introduction Perturb I Perturb II Beating random Conclusion
Arithmetic progressions
3AP: Using 12 blocks we have 85.4% of random with γ=1172192
.
4AP: Using 36 blocks we have 82.6% of random with
γ = 1793962930221810091247020524013365938030467437975104177418768222598213753754515890676996254443021344 .
5AP: Using 117 blocks we have 73.2% of random with
γ = 32168(···225 digits··· )878095624321(···225 digits··· )51792 .
Introduction Perturb I Perturb II Beating random Conclusion
We have been able to use our technique to find many coloringthat beat random. But we have not proven that any of thesecolorings are “best”.For 3APs the best known lower bound is due toParrilo-Robertson-Saracino
167532768
= 0.0511169 . . . ≤ γ ≤ 1172192
= 0.0533759 . . . .
Related to the following geometric problem:Subdivide [0,1] into finitely many blocks onthe x-axis, use the same subdivision for [0,1]on the y -axis and create a checkerboard pat-tern (blue in the lower-left corner). Maximizethe amount of red inside the triangle with ver-tices at (0,0), (0,1) and (1,1/2).
Best known pattern is a scaled version of 28-6-28-37-59-116.
Introduction Perturb I Perturb II Beating random Conclusion
Open problems
Why is symmetry (and in particular anti-symmetry) socommon?Why does the constellation pattern {0, 2
5 ,1} go toalternating blocks? Which other constellations should alsohave that behavior?Show that any constellation pattern with 4 or more pointshas a coloring that beats random.Find a coloring that ties random for k -term arithmeticprogressions.What happens when we allow for more than two colors?
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