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Results tagged with combinatorial-designs
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user 8008
Design theory is the subfield of combinatorics concerning the existence and construction of highly symmetric arrangements. Finite projective planes, latin squares, and Steiner triple systems are examples of designs.
4
votes
colorings of ${\mathbb Z}^d$ with constraints
A periodic $(n,k)$ coloring can be modified to give a non-periodic $(mn+\ell,mk+\ell)$ coloring for any non-negative $m,\ell$ not both $0$ (this uses some comments to improve my previous answer, altho …
- 30.1k
7
votes
Accepted
How many elements with a hamming distance of 3 or less?
You are not using the positions at all. You have 50 points. $S$ is the set of all $\binom{50}{5}=2118760$ selections of 5 points. You want a subset $B \subset S$ such that any $s \in S$ intersects at …
4
votes
Meeting management
Look at the edit history for an old idea (if you wish).
Here is a schedule which runs for 8 days. Each row is an agenda for a day. It consists of 60 digits in 6 groups of 10 (for readability). Note t …
- 30.1k
1
vote
Accepted
Number of points in an intersecting linear hypergraph
The usual proof of Fisher's inequality is very finite. I will call things in $\mathcal{P}$ points and those in $\mathcal{L}$ lines although the reverse might be more natural. Let $L=|\mathcal{L}|,P=| …
- 30.1k
2
votes
Pairwise balanced designs with $r=\lambda^{2}$
With blocks allowed to repeat and block sizes allowed to differ, you should be able to do lots of things if you do not insist that the number of points be small. I am about to add another answer with …
- 30.1k
1
vote
Pairwise balanced designs with $r=\lambda^{2}$
Here is a perspective different enough to merit being its own answer. Perhaps it is already familiar to you.
I find it easier to think of the equivalent dual problem: Look instead for designs where …
- 30.1k
2
votes
Known result about existence of $n$-vertex $k$-uniform $r$-hypergraphs?
I will, as is common, use $v$ rather than $n$ for $|V|.$ The number of edges (often called blocks) is $b=\frac{vr}k$ and this must be an integer.
If you are motivated by finite projective planes and …
- 30.1k
5
votes
Best strategy for a combinatorial game
The question is a little ambiguous as to what it is asking. But the idea is clear and it is a nice question. As I interpret it, I think the answer might be no and that designs and finite geometries mi …
- 30.1k
0
votes
Best strategy for a combinatorial game
I gave a longer answer before you clarified the problem. Here is a briefer one. I will mainly talk about $N=100$ and $k=20.$
Note that choosing $\frac{N}k$ groups the strategy and optimum outcome are …
- 30.1k
12
votes
Is there a 7-regular graph on 50 vertices with girth 5? What about 57-regular on 3250 vertices?
Additional random facts.
The Peterson Graph can be obtained by identifying the antipodal points of a dodecahedron and it has $S_5$ as its automorphism group (order 120 of course).
There are a numbe …
- 30.1k