Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Generalized Kings and Single-Elimination Winners in Random Tournaments   Tournaments can be used to model a variety of practical scenarios including
sports competitions and elections. A natural notion of strength of alternatives
in a tournament is a generalized king: an alternative is said to be a $k$-king
if it can reach every other alternative in the tournament via a directed path
of length at most $k$. In this paper, we provide an almost complete
characterization of the probability threshold such that all, a large number, or
a small number of alternatives are $k$-kings with high probability in two
random models. We show that, perhaps surprisingly, all changes in the threshold
occur in the range of constant $k$, with the biggest change being between $k=2$
and $k=3$. In addition, we establish an asymptotically tight bound on the
probability threshold for which all alternatives are likely able to win a
single-elimination tournament under some bracket.

31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int