5,601 research outputs found
Minimal Stable Sets in Tournaments
We propose a systematic methodology for defining tournament solutions as
extensions of maximality. The central concepts of this methodology are maximal
qualified subsets and minimal stable sets. We thus obtain an infinite hierarchy
of tournament solutions, which encompasses the top cycle, the uncovered set,
the Banks set, the minimal covering set, the tournament equilibrium set, the
Copeland set, and the bipartisan set. Moreover, the hierarchy includes a new
tournament solution, the minimal extending set, which is conjectured to refine
both the minimal covering set and the Banks set.Comment: 29 pages, 4 figures, changed conten
Dominating the Erdos-Moser theorem in reverse mathematics
The Erdos-Moser theorem (EM) states that every infinite tournament has an
infinite transitive subtournament. This principle plays an important role in
the understanding of the computational strength of Ramsey's theorem for pairs
(RT^2_2) by providing an alternate proof of RT^2_2 in terms of EM and the
ascending descending sequence principle (ADS). In this paper, we study the
computational weakness of EM and construct a standard model (omega-model) of
simultaneously EM, weak K\"onig's lemma and the cohesiveness principle, which
is not a model of the atomic model theorem. This separation answers a question
of Hirschfeldt, Shore and Slaman, and shows that the weakness of the
Erdos-Moser theorem goes beyond the separation of EM from ADS proven by Lerman,
Solomon and Towsner.Comment: 36 page
Gamma-Set Domination Graphs. I: Complete Biorientations of \u3cem\u3eq-\u3c/em\u3eExtended Stars and Wounded Spider Graphs
The domination number of a graph G, γ(G), and the domination graph of a digraph D, dom(D) are integrated in this paper. The γ-set domination graph of the complete biorientation of a graph G, domγ(G) is created. All γ-sets of specific trees T are found, and dom-γ(T) is characterized for those classes
Some relational structures with polynomial growth and their associated algebras II: Finite generation
The profile of a relational structure is the function which
counts for every integer the number, possibly infinite, of
substructures of induced on the -element subsets, isomorphic
substructures being identified. If takes only finite values, this
is the Hilbert function of a graded algebra associated with , the age
algebra , introduced by P.~J.~Cameron.
In a previous paper, we studied the relationship between the properties of a
relational structure and those of their algebra, particularly when the
relational structure admits a finite monomorphic decomposition. This
setting still encompasses well-studied graded commutative algebras like
invariant rings of finite permutation groups, or the rings of quasi-symmetric
polynomials.
In this paper, we investigate how far the well know algebraic properties of
those rings extend to age algebras. The main result is a combinatorial
characterization of when the age algebra is finitely generated. In the special
case of tournaments, we show that the age algebra is finitely generated if and
only if the profile is bounded. We explore the Cohen-Macaulay property in the
special case of invariants of permutation groupoids. Finally, we exhibit
sufficient conditions on the relational structure that make naturally the age
algebra into a Hopf algebra.Comment: 27 pages; submitte
Set-Rationalizable Choice and Self-Stability
A common assumption in modern microeconomic theory is that choice should be
rationalizable via a binary preference relation, which \citeauthor{Sen71a}
showed to be equivalent to two consistency conditions, namely
(contraction) and (expansion). Within the context of \emph{social}
choice, however, rationalizability and similar notions of consistency have
proved to be highly problematic, as witnessed by a range of impossibility
results, among which Arrow's is the most prominent. Since choice functions
select \emph{sets} of alternatives rather than single alternatives, we propose
to rationalize choice functions by preference relations over sets
(set-rationalizability). We also introduce two consistency conditions,
and , which are defined in analogy to and
, and find that a choice function is set-rationalizable if and only if
it satisfies . Moreover, a choice function satisfies
and if and only if it is \emph{self-stable}, a new concept based
on earlier work by \citeauthor{Dutt88a}. The class of self-stable social choice
functions contains a number of appealing Condorcet extensions such as the
minimal covering set and the essential set.Comment: 20 pages, 2 figure, changed conten
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