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.
Suppose that a binary operation $$\circ $$ ∘ on a finite set X is injective in each variable separately and also associative. It is easy to prove that $$(X,\circ )$$ ( X , ∘ ) must be a group. In this paper we examine what happens if one knows only that a positive proportion of the triples $$(x,y,z)\in X^3$$ ( x , y , z ) ∈ X 3 satisfy the equation $$x\circ (y\circ z)=(x\circ y)\circ z$$ x ∘ ( y ∘ z ) = ( x ∘ y ) ∘ z . Other results in additive combinatorics would lead one to expect that there must be an underlying ‘group-like’ structure that is responsible for the large number of associative triples. We prove that this is indeed the case: there must be a proportional-sized subset of the multiplication table that approximately agrees with part of the multiplication table of a metric group. A recent result of Green shows that this metric approximation is necessary: it is not always possible to obtain a proportional-sized subset that agrees with part of the multiplication table of a group.
We prove a number of results related to a problem of Po-Shen Loh [9], which is equivalent to a problem in Ramsey theory. Let a = (a1, a2, a3) and b = (b1, b2, b3) be two triples of integers. Define a to be 2-less than b if a i < b i for at least two values of i, and define a sequence a1, …, a m of triples to be 2-increasing if a r is 2-less than a s whenever r < s. Loh asks how long a 2-increasing sequence can be if all the triples take values in {1, 2, …, n}, and gives a log* improvement over the trivial upper bound of n2 by using the triangle removal lemma. In the other direction, a simple construction gives a lower bound of n3/2. We look at this problem and a collection of generalizations, improving some of the known bounds, pointing out connections to other well-known problems in extremal combinatorics, and asking a number of further questions.
We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some k, or the entire multiplication table of a certain large abelian group, as a subgrid. As a consequence, we show that triples systems coming from a finite group contain configurations with t triples spanning O( √ t) vertices, which is the best possible up to the implied constant. We confirm that for all t we can find a collection of t triples spanning at most t + 3 vertices, resolving the Brown-Erdős-Sós conjecture in this context. The proof applies well-known arithmetic results including the multidimensional versions of Szemerédi's theorem and the density Hales-Jewett theorem.This result was discovered simultaneously and independently by Nenadov, Sudakov and Tyomkyn [5], and a weaker result avoiding the arithmetic machinery was obtained independently by Wong [11].
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