3 dimensional matching shortest solution NP-hard? We have array of arbitrary number of elements - 3d vectors with positive integers components - for example

let a=[ [0,1,2], [1,0,2], [1,1,1], [1,2,0 ], [2,0,1 ], [2,1,0 ] ];


And we want to remove elements from that list, which have duplicate value on i-th index with other elements. This problem can have more than one solution:


solution with 3 elements: [0,1,2],[1,2,0],[2,0,1]
solution with 2 elements: [1,0,2],[2,1,0]


As you can see, the solution has this property that each solution element have unique value on i-th index (numbers on i-th position never duplicate) and if we add any other element from array a to that solution we loose this property.

Optimalization: from this wiki aricle I know that finding longest solution is NP-hard problem - but what about finding the shortest solution? Does exists algorithm which not use brute-force to find shortest solution (shortest 3-d matching)?

Here is interactive visualisation in javascript (works in browser, I put it also in wiki article). Here is implementation for find any solution in javascript.
Your problem is NP-complete, even in two dimensions. 
There is a straightforward reduction from MINIMUM MAXIMAL MATCHING in bipartite graphs:


  MINIMUM MAXIMAL MATCHING in bipartite graphs
  INSTANCE: a bipartite graph $G=(V_1\cup V_2,E)$ with $E\subseteq V_1\times V_2$; an integer $k$
  QUESTION: Does $G$ possess a maximal matching $E'$ of cardinality at most $k$?


(A maximal matching is a subset $E'\subseteq E$ such that no two edges in $E'$ share a common endpoint, and such that every edge in $E-E'$ shares a common endpoint with some edge in $E'$.)

NP-hardness of MINIMUM MAXIMAL MATCHING in bipartite graphs has been shown in 


  M. Yannakakis, F. Gavril
  Edge dominating sets in graphs.
  SIAM J. Appl. Math. 38, 364–372 (1980)
  https://epubs.siam.org/doi/10.1137/0138030


For the reduction, take the vertex set $V_1\cup V_2$ as ground set.
For every edge $\{u,v\}\in E$ with $u\in V_1$ and $v\in V_2$, create a corresponding vector $[u,v]$.
The equivalence to your problem is immediate.

(And if you insist on 3-dimensional vectors, then you should add a third component to these vectors and fill it with a lot of different values.)