Let $ \mathcal{T}= \{A \subset \{1,2,\ldots,9 \} \ ; \ |A|=5 \}$. Find $n_{\min}$... 
Let $ S=\{1,2,\ldots,9 \}$ and $ \mathcal{T}= \{A \subset S \ ; \ |A|=5 \}$.
  Find the minimum value of $n$ such that for any $ \mathcal{U} \subset \mathcal{T} $ with $|\mathcal{U}|=n$ there exist two sets $A,B \in \mathcal{U}$ so that $ |A \cap B|=4$.

Let $\mathcal{F}=\{A_1,A_2,\ldots,A_k\} \subset \mathcal{T}$ be a family of $5$-element subsets of $S$ such that: $$|A\cap B|\leq 3\;\;\;\;\;\;\forall A,B \in \mathcal{F}, A\neq B$$ Now we are interested in $k_{\max}$.
Make a following bipartite graph. Connect $A_i$ with a $4$-element subset $B\subset S$ if and only if $B\subset A_i$.
Then the degree of any $B$ is at most $1$, while the degree of each $A_i$ is $ {5\choose 4}=5$.
So we have $1\cdot {9\choose 4}\geq 5 \cdot k$, and so $k\leq 25$. Thus the partial answer is  $n_{min}\leq 26$.
Now I don't know how to find a configuration for $n=26$. If I understand we are searching for Steiner system $S(4,5,9)$?
 A: The question admits two interpretations from well-studied topics, see the references.    
The first is combinatorial, concerning $A(n, d, w)$, which is the maximal possible  number of binary vectors of  length $n$, (Hamming) distance at least $d$ apart, and constant weight (that is, the number of $1’$s) $w$. It is also related with $A(n, d)$, the maximal possible  number of binary vectors of  length $n$ and distance at least $d$ apart (with  no  restriction on weight). Now we see that $k_{\max}=A(9,4,5)$. This value (18) was known already in 1990, see, for instance, [BSSS, Table I-A]. Before I found this reference, my program based on a random choice computed a few maximal systems of size 18 of binary vectors of length $9$, weight $5$, and distance at least $4$ between any two distinct vectors, see below. Another proof of their optimality follows from an observation that $A(9,4)=20$ [Be] and that when we add to any such system binary vectors $(0,\dots,0)$ and $(1,\dots,1)$ we obtain a system of $20$ binary vectors of  length $9$ and distance at least $4$ apart. Remark that  Hougardy has found that there are only two non-equivalent even $(9,4)$  codes of the maximal size $20$ ([Z, p.10]).
The second topic belongs to graph theory considering so-called uniform subset graphs. Namely, a uniform subset graph  $G(n,k,r)$ has a vertex-set consisting of all $k$-element subsets of the set $[n]$ and any two vertices $v$ and $u$ of $G$ are adjacent iff $|v\cap u|=r$. Then $k_{\max}=\alpha(G(9,5,4))$ is the independence number of the graph $G(9,5,4)$. This is a special case of a 
uniform subset graphs for which $r=k-1$. It is called a Johnson graph $J(n,k)$. According to [ACLR], “The Johnson graphs have been studied from several approaches, see for example [A, R, T]. In particular, the determination of the exact value of the independence number $\alpha(J(n, k))$ of the Johnson graph, as far as we know, remains open in its generality, albeit it has been widely studied [Br, BE$_2$, BE, J, MPP]”. 
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References
[A] S. H. Alavi, A generalization of Johnson graphs with an application to triple factorisations, Discrete Math. 338:11 (2015), 2026-2036.
[ACLR] Hernán de Alba, Walter Carballosa, Jesús Leaños, Luis Manuel Rivera, Independence and matching numbers of some token graphs.
[Be] M. R. Best, Optimal codes, in Packing and Covering in Combinatorics, ed. A. Schrjiver, Mathematical Centre Tracts 106 (1979), 119-140.
[Br] A. E. Brouwer, Bounds on A(n, 4,w).
[BE] S. Bitan, T. Etzion,  On the chromatic number, colorings, and codes of the Johnson graph, Discrete Appl. Math. 70:2 (1996), 163-175.
[BE$_2$] A. E. Brouwer, T. Etzion, Some new distance-4 constant weight codes, Adv. Math. Commun., 5:3 (2011), 417-424.
[BSSS] A. E. Brouwer, Lames B. Shearer, N. I. A. Sloane, Warren D. Smith, A new table of constant weight codes, IEEE Trans. Inform. Theory., 36:6 (1990), 1335-1380.
[E] Joakim Ekberg, Geometries of Binary Constant Weight Codes, Master thesis, Karlstadt Universitet, Faculty 2 Department of Mathematics, 2006. 
[EV] Tuvi Etzion, Alexander Vardy, A New Construction for Constant Weight Codes.
[J] S. M. Johnson, A new upper bound for error-correcting codes, IRE Trans. Inform. Theory 8:3 (1962), 203-207.
[MPP] K. G. Mirajkar, K. G. and Y. B. Priyanka, Traversability and Covering Invariants of Token Graphs, Mathematical Combinatorics 2 (2016), 132-138.
[R] H. Riyono, Hamiltonicity of the graph g(n; k) of the Johnson scheme, Jurnal Informatika 3 (2007), 41-47.
[T] P. Terwilliger, The Johnson graph $J(d, r)$ is unique if $(d, r)\ne (2; 8)$, Discrete Math. 58:2 (1986), 175-189.
[Z] Günter M. Ziegler. Coloring Hamming Graphs, Optimal Binary Codes, and the 0/1-Borsuk Problem in Low Dimensions. 
A: Continuing the thread in my comment above, I did a computer search for a counterexample using this Python implementation of Algorithm X. I found no solutions. You can run the program yourself here. To help convince you I coded everything correctly, I used the same algorithm to find $S(5,6,12)$ by brute force. You can play with the code to see it find/not find $S(k-1,k,n)$ for other values of $k,n$, and verify this agrees with the (non)existence of Steiner $k$-tuple systems for your chosen $n$.
In short, I think $n_{\text{min}}\le 25$. 
