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Results tagged with combinatorial-designs
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user 7076
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.
8
votes
When do such regular set systems exist?
Switching to complements, the question is if we can choose 77 6-subsets of an 11-set $M$ such that any 5-subset of $M$ is contained in a chosen 6-subset (it is clear that this subset would be unique b …
- 34.3k
3
votes
Accepted
On the existence of symmetric matrices with prescribed number of 1's on each row
Replace each $-1$ with $0$. Then the matrix you look for is the adjacency matrix of a graph with a given degree sequence $r_1, r_2, \dots$. Reconstruction of such a graph (and its adjacency matrix) is …
- 34.3k
3
votes
Accepted
One question about nega-cyclic Hadamard matrices
Such matrices do not exist as from the parity consideration already first two rows cannot be orthogonal.
- 34.3k
5
votes
Accepted
Solving a Diophantine equation related to Algebraic Geometry, Steiner systems and $q$-binomi...
Let $t = 1+q+q^2+\dots+q^n $ then each of the equations (1) and (2) implies that $24t+1$ is a square (namely, $24t+1=(12k+1)^2$ and $24t+1=(12k+5)^2$, respectively). For $n=2$ that leads to a Pellian …
- 34.3k
2
votes
Does an $(x, bx)$-biregular graph always contain a $x$-regular bipartite subgraph?
The claim does not hold. Here is a counterexample with $a=7$, $b=2$ and $x=3$.
This graph has partite sets of sizes $|U|=14$ and $|V|=7$. Labeling vertices of $V$ as $1,2,\dots,7$, the vertices in $U$ …
- 34.3k