Given two finite posets $\mathcal P$ and $\mathcal Q$, their Ramsey number, denoted by $R(\mathcal P,\mathcal Q)$, is defined to be the smallest integer $N$ such that any blue/red colouring of the vertices of the hypercube $Q_N$ has either a blue induced copy of $\mathcal P$, or a red induced copy of $\mathcal Q$. Axenovich and Walzer showed that, for fixed $\mathcal P$, $R(\mathcal P, Q_n)$ grows linearly with $n$. However, for the diagonal question, we do not even come close to knowing the order of growth of $R(Q_n,Q_n)$. The current upper bound is $R(Q_n,Q_n)\leq n^2-(1-o(1))n\log n$, due to Axenovich and Winter. What about lower bounds? It is trivial to see that $2n\leq R(Q_n,Q_n)$, but surprisingly, even an incremental improvement required significant work. Recently, an elegant probabilistic argument of Winter gave that, for large enough $n$, $R(Q_n,Q_n)\geq 2.02n$. In this paper we show that $R(Q_n,Q_n)\geq 2.7n+k$, where $k$ is a constant. Our current techniques might in principle show that in fact, for every $ε>0$, for large enough $n$, $R(Q_n,Q_n)\geq (3-ε)n$. Our methods exploit careful modifications of layered-colourings, for a large number of layers. These modifications are stronger than previous arguments as they are more constructive, rather than purely probabilistic.
Given two finite posets, P and Q, we denote by R(P, Q) the minimum integer N such that any blue/red copy of the hypercube Q N contains either a blue induced copy of P, or a red induced copy of Q. R(P, Q) is known as the Ramsey number of posets P and Q.
Since any finite poset can be embedded in a hypercube, one of the most natural questions is: what is R(Q m , Q n )? When m is fixed and n grows, these are known as the off-diagonal Ramsey numbers for posets, and R(Q n , Q n ) is known as the diagonal Ramsey numbers for posets. These notions were first studied by Axenovich and Walzer in 2017 [1]. They also established the first bounds, namely that for any positive integers m, n we have
This already shows that the off-diagonal Ramsey numbers have linear growth. However, it places R(Q n , Q n ) between 2n and n 2 + 2n, and there is much desire to close the gap between these bounds.
The most recent upper bound is due to Axenovich and Winter [2] who showed that
On the other hand, the lower bound is notoriously more elusive. While 2n is trivially achieved by colouring the sets of Q 2n-1 blue if the size is at most n -1, and red otherwise (there is no monochromatic chain on length n + 1, let alone an entire Q n ), any little improvement took time, clever optics and new techniques. First, Cox and Stolee improved the upper bound to 2n + 1 for n ≥ 13 and 3 ≤ n ≤ 8 [4]. Their colouring still was blue up to a certain level, and red after a certain level, except that in-between they managed to add one extra level using probabilistic methods. Their bound was later achieved by Bohman and Peng for all n ≥ 3, this time by an explicit colouring [3]. Moreover, Grósz, Methuku and Thompkins showed that, in fact, for any positive integers n, m, R(Q m , Q n ) ≥ m + n + 1, which is surprisingly still the best lower bound for the off-diagonal case [5].
Recently, using clever probabilistic methods, Winter [6] achieved the best lower bound yet for diagonal Ramsey numbers for posets, namely that for large enough n,
In this paper we show the following.
Theorem 1. For n large enough, there exists a constant k such that R(Q n , Q n ) > 2.7n + k.
Our blueprint for this result can, in principle, give a lower bound of (3 -ϵ)n for every fixed ϵ > 0 and n large enough. We explore the absolute limitations of this method at the end of the paper.
Before explaining the strategy, we introduce some standard terminology. For a natural number n, we define [n] = {1, 2, . . . , n}. The n-dimensional hypercube, denoted by Q n , is the poset consisting of all subsets of [n] ordered by inclusion. An embedding of a poset P into a poset Q is an injective map ϕ : P → Q that preserves the poset structure -that is, x ≤ P y if and only if ϕ(x) ≤ Q ϕ(y). For 0 ≤ i ≤ N , we refer to the collection of subsets of [N ] of size i, denoted by [N ] (i) , as the i th level of Q N . A colouring of Q N is called layered if sets of the same size have the same colour. The number of layers is the number of colour transitions, plus one, when going from the empty set to the full set. A layer is a maximal collection of monochromatic levels of Q N .
Our proof is inspired by Winter’s proof in which a 4-layered colouring is modified with the help of two families of sets, called pivots. Our strategy is to start with a large number of layers, pair them up, and inside each pair modify the colouring according to two families of pivots tailored for said pair. The pivots (and their parameters) are designed to force a monochromatic embedding to ‘skip’ a certain number of levels. If one manages to force a monochromatic embedding to skip more than n + 1 levels in total, there is, of course, no monochromatic Q n . In Winter’s proof, the existence of pivots relies exclusively on probabilistic methods. Our proof is more constructive in the sense that we pinpoint exactly one type of families of pivots. This gives more control and, in consequently, tighter bounds. It turns out that starting with 602 layers is enough to achieve the claimed 2.7n bound. Moreover, for an arbitrary number of initial layers, we pinpoint the exact restrictions that need to be satisfied in order for the proof to go through, which suggest that the more layers, the better the bound, although not exceeding 3n.
The plan of the paper is as follows. In Section 2, to illustrate our methodology without overly complex parameters, we establish a weaker lower bound of (2 + 1/3)n, assuming the existence of pivot sets, for a 6-layered initial colouring. In Section 3, the most technical part of the paper, we rigorously prove the existence of the necessary pivot sets (Lemma 4 and Lemma 5). Finally, in Section 4, we present the proof of our main result, for which the parameters were found using a numerical optimiser (see Appendix), as well as establish the absolute technical limitations of this type of proof strategy.
The proof is heavily based on the existence of certain sets of ‘pivots’. We first state the lemmas that ensu
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