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The kind of object you're looking at is exactly an $(r, \lambda)$-design for $\lambda = 1$ in combinatorial design theory. An $(r, \lambda)$-design is an ordered pair $(V, \mathcal{B})$, where $\math …

No. You can't improve it to $o(n^2)$. Let $\operatorname{ex}_{C_2}(n)$ be the largest possible number of edges of a $4$-uniform hypergraph on $n$ vertices that contains no cycle of length $2$. You ca …

As mentioned in Hao Chen's answer, what you're looking for seems to be a good mixed code. There don't seem to be many papers on this. But apparently the following paper gives the best known general up …

To give an idea of how real-life applications may arise outside mathematics, let's consider a "testing" problem of some sort. I'll set up a problem to solve first, so Sperner's theorem doesn't appear …

Sounds like similar to a forbidden configuration problem in extremal set theory, hypergraph theory, and design theory. I don't know if exactly the same problem has been studied in one of those fields …

The fact that every odd dimensional projective geometry ${\rm PG}(n,2)$ over $\mathbb{F}_2$ admits a line parallelism (i.e., the Steiner $2$-design formed by the points and lines of ${\rm PG}(n,2)$ wi …

I found this paper I. M. Gessel, G. Xin, A Combinatorial Interpretation of the Numbers $6(2n!)/n!(n+2)!$, Journal of Integer Sequences 8 (2005) Article 05.2.3 whose abstract says: It is well kno …

I am interested in a sharp bound on the largest possible size $e_3({\boldsymbol{Z}_n})$ of a subset $S \subset \boldsymbol{Z}_n$ such that for any three distinct elements $a, b, c \in S$ we have $a+b …

You're asking what the number of blocks of a maximum packing is. An ordered pair $(S, \mathcal{B})$ of a finite set $S$ of cardinality $\vert S \vert = v$ and a finite set $\mathcal{B}$ of $k$-subset …

Unfortunately, no. It is known that the maximum number of mutually disjoint $S(4,5,11)$s on the same point set is $2$. Any such pair are always isomorphic. So, you can't find $7$ disjoint copies of an …

I think you're assuming $x \not= y$ when you say "for any $x, y \in S$." In any case, your question seems like a mix of coding theory and design theory. If you find the case when $a = b = \frac{n}{2} …

My previous answer is already too long, and this is too much to include in a comment. But I found a paper that studies the problem you asked here: R. M. Roth, G. Seroussi, Bounds for binary codes wit …

It is called perfect hash families in the design theory and computer science literature. A perfect hash family PHF$(N; k, v, t)$ is an $N \times k$ array on $v$ symbols with $v \geq t$, where for eve …

Edit: The possible "gap" of sort in Caro and Yuster's proof of their upper bound has just been fixed! See Ben Barber's comment below (and his joint paper with Daniela Kühn, Allan Lo and Deryk Osthus o …

I think it is an optimal version of a covering with index $1$. A $t$-$(n,k,\lambda)$ covering is an ordered pair $(U,\mathcal{B})$ of a finite set $U$ of cardinality $n$ and a finite set $\mathcal{B} …