# 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 family $\mathcal F$ of size $25$ would not quite be a Steiner system, as the number of subsets of $\{1,\dots,9\}$ of size four contained in some element of $\mathcal F$ would be $5\cdot 25$, which is one less than $\binom94=126$. – Mike Earnest Jan 16 at 20:51
• You want $25$ five-element sets whose four-element subsets comprise all but one of the four element subsets of $[9]$. Consider the $120\times 125$ matrix of 0's and 1's whose columns are indexed by four-element subset of $[9]$, except for $\{1,2,3,4\}$, and whose rows are indexed by five-element subsets of $[9]$, except for those who contain $\{1,2,3,4\}$. Then you want a subset of rows whose sum is the vector of $125$ ones, with a one where the row contains the column. This is solvable by Knuth's Algorithm X. – Mike Earnest Jan 16 at 21:05
• @MikeEarnest There are ${9\choose 5}-|\{5,6,7,8,9\}|=126-5=121\ne 120$ five-element subsets of $[9]$, except for those who contain $\{1,2,3,4\}$. – Alex Ravsky May 7 at 4:00

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.

• +1 This was worth the wait. :D – antkam May 8 at 5:11

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 $$n_{\text{min}}\le 25$$.