Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some results on the Ryser design conjecture-III   A Ryser design $\mathcal{D}$ on $v$ points is a collection of $v$ proper
subsets (called blocks) of a point-set with $v$ points such that every two
blocks intersect each other in $\lambda$ points (and $\lambda < v$ is a fixed
number) and there are at least two block sizes. A design $\mathcal{D}$ is
called a symmetric design, if every point of $\mathcal{D}$ has the same
replication number (or equivalently, all the blocks have the same size) and
every two blocks intersect each other in $\lambda$ points. The only known
construction of a Ryser design is via block complementation of a symmetric
design. Such a Ryser design is called a Ryser design of Type-1. This is the
ground for the Ryser-Woodall conjecture: "every Ryser design is of Type-1".
This long standing conjecture has been shown to be valid in many situations.
  Let $\mathcal{D}$ denote a Ryser design of order $v$, index $\lambda$ and
replication numbers $r_1,r_2$. Let $e_i$ denote the number of points of
$\mathcal{D}$ with replication number $r_i$ (with $i = 1, 2$). Call a block $A$
of $\mathcal{D}$ small (respectively large) if $|A| < 2\lambda$ (respectively
$|A| > 2\lambda$) and average if $|A|=2\lambda$. Let $D$ denote the integer
$e_1 - r_2$ and let $\rho> 1$ denote the rational number
$\dfrac{r_1-1}{r_2-1}$. Main results of the present article are the following:
An equivalence relation on the set of Ryser designs is established. Some
observations on the block complementation procedure of Ryser-Woodall are made.
It is shown that a Ryser design with two block sizes one of which is an average
block size is of Type-1. It is also shown that, under the assumption that large
and small blocks do not coexist in any Ryser design equivalent to a given Ryser
design, the given Ryser design must be of Type-1.

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The categories of this paper are: hep-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: astro-ph
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The categories of this paper are: adap-org
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The categories of this paper are: physics
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom