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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 …

One of the more convenient and popular approximations of the sum is $$\frac{2^{nH(\frac{k}{n})}}{\sqrt{8k(1-\frac{k}{n})}} \leq \sum_{i=0}^k\binom{n}{i} \leq 2^{nH(\frac{k}{n})}$$ for $0< k < \frac{ …

Edit: Since this just became available on arXiv, Peter Keevash solved the existence conjecture of Steiner $t$-designs, which means that what I wrote below a year ago as one of the most important open …

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 …

To the eye of younger folks like me who doesn't know or care exactly why JCT split into two, Series B looks like a specialized journal almost entirely in graph theory while Series A deals with a broad …

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 …

So, the supposedly the sharpest one among all known bounds is somehow poorer than the bound you learn in Coding Theory 101 if the alphabet size $q$ approaches infinity. I think the reason you find it …

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} …

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 …

To complement Timothy Chow's nice answer, here's a recent and great survey on this topic if anyone is interested: D. S. Stones, The Many Formulae for the Number of Latin Rectangles, Electron. J. Comb …

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 …

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 …

Lucia's answer given in the linked question from the comment gives an upper bound (i.e., no pair should appear twice as a subset of $A_i$). But of course, the real question starts from here: When can …

An elementary counting argument shows that $2$-$(v,3,3)$ exists only if $v$ is odd (or, more precisely, for $\lambda \equiv 3 \pmod{6}$ a $2$-$(v,3,\lambda)$ exists only if $v \equiv 1 \pmod{2}$). Thi …

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 …