# Tagged Questions

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### Can we sometimes define the parity of a set?

Suppose that ${n\choose k}, {n-1\choose k-1}, \ldots, {n-k+1\choose 1}$ are all even. (This happens for example if $k=2^\alpha-1$ and $n=2k$.) In this case, can we select ${n\choose k}/2$ sets of size ...
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### Hitting sets (aka covers aka transversals) of Steiner triple systems

Does there exist a constant $c$ so that the lines of every Steiner triple system on $v$ points can be covered by $cv$ points? That is if $D \in STS(v)$ with point set $T=\{1,2,\ldots,v\}$ then ...
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### Number of blocks in a t-(v,k,l) design with empty intersection with a given set U [closed]

Question Given a $t-(v,k,\lambda)$ design $(X,\mathcal{B})$ and a set $U\subset X$ with $|U|=u\leq t$, what is the number of blocks $B\in\mathcal{B}$ such that $B\cap U=\emptyset$? The answer is: ...
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### Status of Hadamard matrix conjecture

I would like to know if any progress has been made on Hadamard conjecture : There exists a Hadamard matrix of order $n=4k$ $\forall k \in \mathbb{N}$.
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### The Symmetry of Steiner System S(5,8,24)

The group of automorphisms of S(5,8,24), M_{24}, is 5-transitive. Other than Symmetric groups are there any other 5-transitive groups? If not, would it be correct to say S(5,8,24) is the most ...
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### Constructing Steiner Triple Systems Algorithmically

I want to create STS(n) algorithmically. I know there are STS(n)s for $n \cong 1,3 \mod 6$. But it is difficult to actually construct the triples. For STS(7) it is pretty easy and but for larger n I ...
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### Lower bounding the maximum size of sets in a set family with union promise

The following problem has come up while working on the relationship between certificate and randomized decision tree complexities of boolean functions. However, I think it is of interest by itself and ...
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### A generalization of covering designs and lottery wheels

This question is inspired by a recent problem . A $(v,k,t)$ covering design is a pair $(V,B)$ where $V$ is a set of $v$ points and $B$ is a family of $k$ point subsets (called blocks) such that ...
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### How many elements with a hamming distance of 3 or less?

[This is a complete rewrite which makes some of the comments redundant or irrelevant.] Take a set of $50$ elements. How many subsets of size $5$ are needed so that every subset of size $5$ will ...
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### Is there a 7-regular graph on 50 vertices with girth 5? What about 57-regular on 3250 vertices?

The following problem is homework of a sort -- but homework I can't do! The following problem is in Problem 1.F in Van Lint and Wilson: Let $G$ be a graph where every vertex has degree $d$. ...
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### What's the maximum determinant of the (0, 1) matrix from M(n, R)?

If there's no exact formula what's the nearest upper and lower bounds do you know?
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### Solving a Diophantine equation related to Algebraic Geometry, Steiner systems and $q$-binomials?

The short version of my question is: 1)For which positive integers $k, n$ is there a solution to the equation $$k(6k+1)=1+q+q^2+\cdots+q^n$$ with $q$ a prime power? 2) For which positive ...
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### Which Steiner systems come from algebraic geometry?

This question is motivated by the ongoing discussion under my answer to this question. I wrote the following there: A $(p, q, r)$ Steiner system is a collection of $q$-element subsets $A$ (called ...