TITLE: Sets of points containing permutations - a Ramsey-type question
QUESTION [16 upvotes]: The following question arised as a side-question in a geometric problem. It has a "feel" similar to problems in Ramsey-theory, but I have not found any mention of it (also I'm not very familiar with the field). Was this problem considered before? Does it have an easy answer?
Consider the set of grid points $[n] \times [n]$, and color each point either black or white, giving rise to the sets $B,W$ (such that $B \cup W = [n] \times [n]$, and $B \cap W = \emptyset$).
Are the following true?

(stronger): Either $B$ or $W$ contains every permutation of $[n/2]$.
(weaker, implied by stronger): Every permutation of $[n/2]$ is contained in either $B$ or $W$.
if true, holds also for $k$ colors and $n/k$? if not true, what is largest $m$ for which it holds?

A set of points $X$ containing a permutation $\sigma$ of $[n]$ means that: there are points $(x_1, y_1), \dots, (x_n,y_n) \in X$, such that $y_1<y_2<y_3<\dots<y_n$, and $x_i$ have the same relative ordering as $\sigma_i$ (meaning: $\sigma_i < \sigma_j  \iff x_i < x_j$, for all $i,j$).
For example, $[n] \times [n]$ contains all permutations of $[n]$.
Some easy observations:

One of $B$ and $W$ might not contain all permutations, even if it contains more than half of the original points (construction: L-shape thinner than n/2).
If $n/2$ bound holds, it is best possible (construction: color left half black, right half white).

REPLY [3 votes]: Here is the sketch of an argument that I heard from Martin Balko. 
It seems to show that the correct asymptotics is $N(k) = k^2 / polylog(k)$. That is, there is a B/W coloring of the $N(k) \times N(k)$ grid such that some permutation of size $k$ is not contained in either the B or the W sets. If true, then this is quite close to the upper bound of Ben Barber from the other answer.
The argument uses a result from the recent paper Ordered Ramsey Numbers by Conlon, Fox, Lee, and Sudakov (http://arxiv.org/abs/1410.5292).
Theorem 2.4 of the paper claims (essentially) the following.
For all $k$, there is some value $N = k^2 / polylog(k)$ and some perfect matching $M$ of a bipartite graph with vertices $L = \{1,...,k\}$, and $R = \{k+1,...,2k\}$, such that we can color B/W the edges of a complete graph $H$ with vertices $\{1,...,N\}$ such that $M$ is not contained either as a B-subgraph, or as a W-subgraph. Here, ``contained'' means that the vertices of $M$ are mapped to vertices of $H$ such as to respect the ordering.
Our problem is essentially the same, with the difference that $H$ is a complete bipartite graph with vertices, say $\{1,...,N/2\}$ and $\{N/2+1,...,N\}$. This can be seen as deleting some of the edges of $H$, or coloring them with some third color that we can not match to anything. Denote the resulting graph as $H'$. If some matching was not contained in $H$, then it is also not contained in $H'$, i.e. by going from the complete to the complete bipartite graph, avoiding some subgraph becomes easier. 
Note: $H'$ plays the role of the matrix, and the ordered matching $M$ plays the role of the contained (or avoided) permutation.
It would be nice to get some intuition how such a coloring looks "visually" -- from the proof in the paper it is hard to get such an intuition.