Dilworth's Lemma 
We want to place $2012$ pockets,
  including variously colored balls,
  into $k$ boxes such that either
  
  
*
  
*For all boxes, all pockets in a box must include a ball with the same
  color 
  
  
  or
  
  
*
  
*For all boxes, all pockets in a box must include a ball having a color
  which is not included in any other
  pockets in this box 
  
  
  Find the smallest value of $k$ for which
  we can always do this placement
  whatever the number of balls in the
  pockets and whatever the colors of
  balls.


I will show that $k>44$.
Let all pockets be monochrome. 
Let  $p_1, p_2, \dots , p_{45}$ have color $c_1$.
Similarly, $p_{46}, p_{47}, \dots , p_{90}$ have color $c_2$. 
$\vdots$
And, $p_{1981}, p_{1982}, \dots, p_{2012}$ have color $c_{45}$.
If you want to satisfy the first condition, you should put all pockets with color $c_i$ into $k_i$. So there should be $45$ boxes.
If you want to satisfy the second condition, you should put each pocket with color $c_i$ into a separate box. So there should be $45$ boxes.
So $k > 44$. But I didn't proved that $k=45$.

I think the problem is related with Dilworth's Lemma or something similar to it. But I couldn't manage to set a proper (reflexive, transitive and antisymmetric) relation to get a partially ordered set to simulate the chains and the anti-chains.

Explanation:
Thinking pockets as sets ($p$), colors as elements ($c$), and boxes as sets of sets ($b$)
$p_i = \{c_{i1}, c_{i2}, \dots \}$
$b_i = \{p_{i1}, p_{i2}, \dots \}$
The problems asks for the least $k$ such that
$b_1 \cup b_2 \cup \dots \cup b_k = \{p_1, p_2, \dots, p_{2012}\}$ where $b_i \cap b_j = \emptyset$, for any $i \neq j$. 
and all $b_i$s (all together) should satisfy one of the statements below:


*

*For each $i$, it should be that $\bigcap\limits_{p \in b_i}p \neq \emptyset$

*For each $i$, and each $j$ where $p_j \in b_i$, it should be that $p_j - \bigcup\limits_{p\in (b_i-p_j)}^{}p \neq \emptyset$ 

I will show that $k\ge1006$. We note $c_i$ the color $i\in\mathbb N$ and if $i\neq j$, $c_i$ and $c_j$ are different colors.
Consider the 1006 pockets $p_i$ ($1\le i\le 1006$) such that $p_i$ contains only one ball of color $c_i$. Under the first condition, we need at least 1006 boxes for those pockets, because no boxes can contain two different $p_i$. 
Consider the 1006 pockets $u_i$ ($1\le i\le 1006$) such that $u_i$ contains only $i$ balls of color $c_k$ for $1\le k\le i$. Under the second condition, we need at least 1006 boxes for those pockets, because no boxes can contain two different $u_i$.
Then the set $\{p_i\}\cup\{u_i\}$ (with 2012 pockets) needs at least 1006 boxes under both conditions.
In fact k=1006 (but I wait for your answer to my question)
