# k-partite design

Is the following true? For every $$n≥1,k≥2n \geq 1, k\geq 2$$, there is a set $$S⊆[n]kS \subseteq [n]^k$$ of size $$|S|=n2|S| = n^2$$ such that every two $$kk$$-tuples in $$SS$$ have at most one common entry.

Does anyone know if this is true? Is there a reference?

Assuming that “at most common entry” means that any two tuples $$(x1,…,xk)(x_1,\ldots,x_k)$$ and $$(y1,…,yk)(y_1,\ldots,y_k)$$ match in at most one position, that is, there is at most one $$ii$$ such that $$xi=yix_i=y_i$$.
The claim is false for $$n=2n=2$$ and any $$k≥4k \ge 4$$. We would need $$n2=4n^2=4$$ tuples $$x,y,z,wx,y,z,w$$. Then each of $$yy$$ and $$zz$$ must differ from $$xx$$ in at least $$k−1k-1$$ positions, which implies that $$yy$$ and $$zz$$ match in at least $$k−2≥2k-2 \ge 2$$ positions.
(If the meaning was that any two tuples contain at most one common value, regardless of position, the claim would fail already with $$n=2n=2$$, $$k=2k=2$$, because there are only four different tuples $$(1,1),(1,2),(2,1),(2,2)(1,1),(1,2),(2,1),(2,2)$$, and $$(1,2),(2,1)(1,2),(2,1)$$ have two common values $$11$$ and $$22$$.)