A New Lower Bound for the Diagonal Poset Ramsey Numbers

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 $R(Q_n,Q_n)\le n^2-(1-o(1))n\log n$, due to Axenovich and Winter. What about lower bounds? It is trivial to see that $2n\le R(Q_n,Q_n)$, but surprisingly, even an incremental improvement required significant work. Recently, an elegant probabilistic argument of Winter gave that, for large enough $n$, $R(Q_n,Q_n)\ge 2.02n$. In this paper we show that $R(Q_n,Q_n)\ge 2.7n+k$, where $k$ is a constant. Our current techniques might in principle show that in fact, for every $\epsilon >0$, for large enough $n$, $R(Q_n,Q_n)\ge (3-\epsilon )n$. 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.