finding patterns avoiding many monochromatic constellations · 2011. 12. 15. · monochromatic...

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

www.math.ucla.edu/~butler


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.


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.)


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?


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


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}.


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.


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.


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)


Starting in blocks

Now we look at runs for the constellation {0, 13 ,1}.

Single block Ten equal blocks


Random starts

Some more runs for the constellation {0, 13 ,1}.

Random Random


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.


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 .


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.


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).


Using the approximate block structure for {0, 13 ,1} we get:

Calculating we have γ =56816777

750000000= 0.07575570 . . ..


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


{∆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.


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 . . ..)


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


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!


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!


Local perturbation run for {0, 25 ,1}


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.


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.


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 .


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.


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?

finding patterns avoiding many monochromatic constellations · 2011. 12. 15. · monochromatic...

Documents