Partition problems in high dimensional boxes

Abstract

Alon, Bohman, Holzman and Kleitman proved that any partition of a d-dimensional discrete box into proper sub-boxes must consist of at least 2 d sub-boxes. Recently, Leader, Milićević and Tan considered the question of how many odd-sized proper boxes are needed to partition a d-dimensional box of odd size, and they asked whether the trivial construction consisting of 3 d boxes is best possible. We show that approximately 2.93 d boxes are enough, and consider some natural generalisations. Question 1.2 (Leader, Milićević, Tan [7]). Let A be a d-dimensional odd box, and let {B 1 , B 2 , . . . , B m } be a partition of A into odd proper sub-boxes. Does it follow that then m ≥ 3 d ?.Our first result is that the answer to this question is 'no': Theorem 1.3. Let d ∈ Z + be divisible by 3. Then there exists a partition of [5] d into 25 d/3 ≤ 2.93 d odd proper sub-boxes.

On 2-colored graphs and partitions of boxes

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We prove that if the edges of a graph G can be colored blue or red in such a way that every vertex belongs to a monochromatic k-clique of each color, then G has at least 4(k − 1) vertices. This confirms a conjecture of Bucic et al [2], and thereby solves the 2-dimensional case of their problem about partitions of discrete boxes with the k-piercing property. We also characterize the case of equality in our result.

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“…In this paper, a 2-colored graph will be a simple graph with edges colored blue or red. Bucic et al [2] asked the following: Given an integer k ≥ 2, what is the smallest possible number of vertices in a 2-colored graph having the property that every vertex belongs to a monochromatic k-clique of each color?…”

Refuting conjectures in extremal combinatorics via linear programming

“…We introduce for each set A ⊂ [n] a 0-1 valued variable x A that indicates whether A ∈ F. We can force F to be an antichain by adding, for each comparable pair A B a linear constraint x A + x B ≤ 1. Next, to ensure that the solution has diameter at most d, for each pair Using an LP solver we attempted to find a counterexample to this conjecture, but according to the computer it holds for the values (n, d) = (10, 3), (8,5), (8,7).…”

“…Example: partitioning a box into proper sub-boxes. Here we describe an application of the methods in the present paper by Bucic, Lidický, Long, and the author [8]. A set of the…”

“…There is a natural partition into 3 d such boxes, and Leader, Milićević and Tan [33] asked whether this is best possible. By phrasing this question as an IP in essentially the same way as above and solving the resulting IP, in [8] we found that [5] vertices.…”