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Results tagged with combinatorial-designs
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user 45255
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.
1
vote
All $2$-designs arising from the action of the affine linear group on the field of prime order
For fixed $k \ge 3$, the generic such design is a full orbit with "index" $\lambda = k(k-1)$ and is probably not of much (combinatorial) interest. Short orbits are, of course, very interesting. As a …
- 1,081
5
votes
What are the major open problems in design theory nowaday?
Personally, I am most interested in design theory with an "asymptotic flavor", and I think there are (edit: were, pre-Keevash) some very interesting open questions in this direction.
To cut to the ch …
4
votes
Accepted
Can we sometimes define the parity of a set?
I wish I had a real answer for you!
You are essentially interested in a tough conjecture of Hartmann, known as the "halving conjecture", which is promoted heavily by Reza Khosrovshahi. Actually, the …
- 1,081
2
votes
Constructions of $2-(v,3,3)$-designs
Yuichiro's is the best answer, but here's another "cheat" answer (which I nonetheless think is kind of neat):
Take a PBD$(v,\{3,5\})$, triple each block of size three, and replace each block of size …
- 1,081
2
votes
Is the domination number of a combinatorial design determined by the design parameters?
Gordon has done a proper search of $(15,3,1)$-designs. I guess my incorrect reasoning does lead to a computer-free proof for (15,3,13)-designs. This is kind of cheating though, because there are rep …
- 1,081
3
votes
Accepted
Hitting sets (aka covers aka transversals) of Steiner triple systems
I think your question equivalently asks if there is a universal constant $c>0$ such that every Steiner triple system of order $v$ has a 'cap' (line-free set) of size at least $c v$. The complement of …
- 1,081
4
votes
Linear algebra proofs in combinatorics?
Variants of the EKR Theorem offer a wide class of examples. This page has a nice list going, by the way.
A friend of mine once made the outrageous claim -- but hear me out -- that most "linear algeb …