Monochromatic Boxes in Colored Grids

Joshua Cooper, USC Math

Steven Fenner, USC CS

Semmy Purewal, College of Charleston Math

“GRID”

“BOX”

][][ 1 daa

},{},{ 11 dd baba

1-D 2-D 3-D

Central Question: For a given grid of dimension d, is it possible to c-color it so thatthere are no monochromatic boxes?

},,1{][ nn

d = 1:

We can c-color [n] without a monochromatic box (i.e., a pair of points) iff n ≤ c.

Definition. Let R = [a1][ad]. If it is possible to c-color it so that there are no

monochromatic boxes, we say that R is c-colorable. Otherwise, R is c-guaranteed.

d = 2:

Suppose c=1. Then [a][b] is c-guaranteed iff a ≥ 2 and b ≥ 2.

Suppose c=2.

Claim: [3][7] is 2-guaranteed, but [3][6] is 2-colorable.

Proof that [3][7] is 2-guaranteed: If any two columns are colored the same,there is a monochromatic rectangle. So, we may assume that the columns represent

all but one of the 8 possible colorings:

Claim: [5][5] is 2-guaranteed, but [4][5] is 2-colorable.

Proof that [5][5] is 2-guaranteed: At least three columns contains at least threereds (up to switching the colors). If there are no monochromatic boxes, then no two oftheir corresponding rows are shared. But, with only five rows, this isn’t possible.

Proof that [5][5] is 2-guaranteed: At least three columns contains at least threereds (up to switching the colors). If there are no monochromatic boxes, then no two oftheir corresponding rows are shared. But, with only five rows, this isn’t possible.

2-guaranteed region

2-colorable region “obstruction set”

For more on 2-d (particularly, 3- and 4-colorability), see forthcoming Fenner,Gasarch, Glover, and Purewal, Rectangle Free Coloring of Grids.

The present work is mostly concerned with what happens in higher d.

These questions can be recast as hypergraph Ramsey problems:

1 2 3 4 5

a

b

c

d

a

b

c

d

1

2

3

4

5

Ramsey version of 2-d problem: Which complete bipartite graphs are c-colorable

without a monochromatic C4?

C4’s govern quasirandomness: The random graph has the fewest copies of C4 for

any graph on a given number of edges, and having close to this number of copiesguarantees all sorts of random-like properties. (See Chung/Graham/Wilson ’89.)

No monochromatic C4 means no monochromatic random graphs, which means the

color classes have to be small. (Already, cn3/2 edges ensures a C4 subgraph.)

A rank 3 tensor with entries in {R,G,B} encodes a complete 3-partite 3-uniformhypergraph:

Ramsey version of general d problem: Which complete d-partite d-uniform

hypergraphs are c-colorable without a monochromatic hyperoctahedron (aka cross-polytope aka orthotope aka dual of the hypercube)?

Hyperoctahedra govern hypergraph quasirandomness! A random d-partite d-uniform

hypergraph has the fewest hyperoctahedra for any hypergraph on a given number of

edges… & having close to this number guarantees all sorts of random-like properties.

Note that the number of boxes in a grid R = [a1][ad] is given by

.2vol22

2

1

2

1

dd

j

jd

j

j Raa

So, define V(c,d) to be the largest integer V so that every d-dimensional grid Rwith volume at most V is c-colorable.

Theorem (CFP).

.22),(

2 2/)1(

2

11

dddd d

c

dcVe d

“Proof”: Lower bound is a straightforward application of Lovász Local Lemma.

(Any given box can only intersect at most 2d vol(R) others, but the

probability of monotonicity is c^(-2^(d-1)).)

Upper bound is a repeated application of Cauchy-Schwarz…

Example 1: How big can n be and still have [3][7][n] be 2-colorable?

Note: If [a][b] is c-colorable, then [a][b][n] is for any n. (Just take fibersof the coloring under projection.)

Claim. [3][7][127] is 2-guaranteed.

Proof. The number of boxes in [3][7] is

.632132

7

2

3

The number of monochromatic boxes is therefore 126. In any 2-coloring of

[3][7][127], each of the [3] [7] “planes” has at least one of these 126monochromatic (2-d) boxes. But then some one is repeated, and such a pair formsa monochromatic 3-d box.

On the other hand, permuting around the rows, columns, and colors of a 2-coloring

of [3] [7] to get all 126 different monochromatic boxes gives a 2-coloring of

[3][7][127].

Example 2: How big can n be and still have [5][5][n] be 2-colorable?

Claim. [5][5][201] is 2-guaranteed.

Proof. Same argument:

.201110212

5

2

52 2

It is still possible to permute around a 2-coloring of [5] [5], but being able to doso depends on the two monochromatic (2-d) boxes occupying disjoint rows/columns.

However…

Claim. [5][5][101] is 2-guaranteed!

Proof. Same argument, only now we observe that any 2-coloring of [5]X[5] actually

admits 2 monochromatic boxes:

.10111012

2

5

2

52

2

So, extend our definition : R is (c,t)-guaranteed if every c-coloring gives rise to tmonochromatic boxes.

Using this idea (and some others, particularly some convex programming), we have

the following upper bounds on n so that [a][b][n] is 2-guaranteed. Mostly

values are known to be within 1 or 2 of the truth.

[3][7][127]

[7][3][127]

[5][5][101]

[22][22][22]

We can iterate the preceding argument to find obstruction sets in dimension d:

[3][7][127] [1008127]

.100812712

127

2

7

2

32

541862 cccccdf

Each exponent is twice thesum of the previous exponents..)(

13

d

cdf

1

2

1

2

12

1

2

1

2

12

1

2

1

ccc

cc

ccc

For general c:

Then the volume is given by

8127931)()(vol cccccdgR2/)13()(

d

cdg

Theorem (CFP). For every element R of the obstruction set,

.vol 2/)13( d

cOR

…which is a lot bigger than c^(2^(d-1)).

This still does not bound the number of obstruction set grids, since the surfacecorresponding to grids of constant volume are infinite. (Although some terrible boundis possible in principle.)

Theorem (CFP). The size of the obstruction set (for d ≥ 3) is

.2/)1317( 3 d

cO c8, c25, c76,…

(The truth is c2 in 2-d.)

More questions…

1. What about if the boxes have to be equilateral? (Direct application to van derWaerden/Szemerédi.)

2. What if we fix c = 2 (say) and let d grow instead?

3. A matching lower bound on the size of the obstruction set.

4. Limiting surface of obstruction set as c ∞? (Even d = 2 is unknown.)

Thank you!