Monochromatic Boxes in Colored Grids

1. Monochromatic Boxes in Colored Grids Joshua Cooper, USC Math Steven Fenner, USC CS Semmy Purewal, College of Charleston Math

3. “GRID” “BOX” 1-D 2-D 3-D Central Question: For a given grid of dimension d, is it possible to c-color it so that there are no monochromatic boxes?

4. d = 1: We can c-color [n] without a monochromatic box (i.e., a pair of points) iff n≤c. d = 2: Suppose c=1. Then [a][b] is c-guaranteediff a≥2 and b≥2. 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. Suppose c=2. Claim:[3][7] is 2-guaranteed, but [3][6] is 2-colorable.

5. 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.

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

7. 2-guaranteed region 2-colorable region “obstruction set” Proof that[5][5] is 2-guaranteed: At least three columns contains at least three reds (up to switching the colors). If there are no monochromatic boxes, then no two of their corresponding rows are shared. But, with only five rows, this isn’t possible. 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.

8. 1 2 3 4 5 a b c d 1 a 2 b 3 c 4 d 5 These questions can be recast as hypergraph Ramsey problems: 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 copies guarantees 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.)

9. A rank 3 tensor with entries in {R,G,B} encodes a complete 3-partite 3-uniform hypergraph: 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 hyperoctahedrafor any hypergraph on a given number of edges… & having close to this number guarantees all sorts of random-like properties.

10. Theorem (CFP). Note that the number of boxes in a grid R = [a1][ad] is given by So, define V(c,d) to be the largest integer V so that every d-dimensional grid R with volume at most V is c-colorable. “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…

11. Proof. The number of boxes in [3][7] is Note: If [a][b] is c-colorable, then [a][b][n] is for any n. (Just take fibers of the coloring under projection.) Example 1: How big can n be and still have [3][7][n] be 2-colorable? Claim. [3][7][127] is 2-guaranteed. 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 126 monochromatic (2-d) boxes. But then some one is repeated, and such a pair forms a 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].

12. Proof. Same argument: 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: Example 2: How big can n be and still have [5][5][n] be 2-colorable? Claim. [5][5][201] is 2-guaranteed. It is still possible to permute around a 2-coloring of [5] [5], but being able to do so depends on the two monochromatic (2-d) boxes occupying disjoint rows/columns.

13. So, extend our definition : R is (c,t)-guaranteed if every c-coloring gives rise to t monochromatic boxes. [7][3][127] [3][7][127] [5][5][101] [22][22][22] 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.

14. [1008127] For general c: Each exponent is twice the sum of the previous exponents. We can iterate the preceding argument to find obstruction sets in dimension d: [3][7][127]

15. Theorem (CFP). For every element R of the obstruction set, Theorem (CFP). The size of the obstruction set (for d ≥ 3) is c8, c25, c76,… Then the volume is given by …which is a lot bigger than c^(2^(d-1)). This still does not bound the number of obstruction set grids, since the surface corresponding to grids of constant volume are infinite. (Althoughsome terrible bound is possible in principle.) (The truth is c2 in 2-d.)

16. More questions… 1. What about if the boxes have to be equilateral? (Direct application to van der Waerden/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!