What is the meaning of "each treatment " in BIBD? My doubt is regarding  Balanced Incomplete Block Design (BIBD)
Consider the following design with 9 treatments and 12 blocks each of size 3, is this design BIBD or not ?

in my opinion it's not BIBD (but I am not sure whether my logic is correct).
This is what my logic is: since we have  9 treatments we must have  $9\choose 2$ treatment pairs. If it would have been BIBD it should include all  $9\choose2$ treatment pairs,  but it does not include all $9\choose2$ treatment pairs, e.g. there is no block having treatment pairs {2,3},{2,5},{3,5},{3,8}, so its not BIBD.
Is my logic correct? (and I can not ask more clear than this)
 A: Yes, your conclusion  and argument is correct.  To make this clearer you could try to draw the graph of the design as in examples of connected designs in DOE  ( you should be able to adapt the code from that post)
Since the design contains $12 \cdot 3 =36$ points, and $\binom{9}{2}=36$, could accomodate a BIBD with the pair cooccurrence constant $\lambda=1$, but some pairs are lacking, as you said, and some, like 28 occurs multiple times. Some examples: 28 in blocks 1,6,9,11,   56 in blocks 5,8 and more. 
To construct a BIBD with your parameters you can try, in R:
library(crossdes)
> D  <-  find.BIB(9,12,3,iter=100)
> D
      [,1] [,2] [,3]
 [1,]    3    6    7
 [2,]    1    5    9
 [3,]    1    2    6
 [4,]    6    8    9
 [5,]    1    7    8
 [6,]    2    4    8
 [7,]    3    5    8
 [8,]    4    5    6
 [9,]    2    3    9
[10,]    2    5    7
[11,]    4    7    9
[12,]    1    3    4
> isGYD(D)

[1] The design is a balanced incomplete block design w.r.t. rows.

For more information, you can have a look at https://www.r-bloggers.com/generating-balanced-incomplete-block-designs-bibd/
