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    A contractive version of a Schur–Horn theorem in II1 factors

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    We prove a contractive version of the Schur–Horn theorem for submajorization in II1 factors that complements some previous results on the Schur–Horn theorem within this context. We obtain a reformulation of a conjecture of Arveson and Kadison regarding a strong version of the Schur–Horn theorem in II1 factors in terms of submajorization and contractive orbits of positive operators.Fil: Argerami, Martin. University of Regina; CanadáFil: Massey, Pedro Gustavo. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; Argentin

    Equistarable graphs and counterexamples to three conjectures on equistable graphs

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    Equistable graphs are graphs admitting positive weights on vertices such that a subset of vertices is a maximal stable set if and only if it is of total weight 11. In 19941994, Mahadev et al.~introduced a subclass of equistable graphs, called strongly equistable graphs, as graphs such that for every c≤1c \le 1 and every non-empty subset TT of vertices that is not a maximal stable set, there exist positive vertex weights such that every maximal stable set is of total weight 11 and the total weight of TT does not equal cc. Mahadev et al. conjectured that every equistable graph is strongly equistable. General partition graphs are the intersection graphs of set systems over a finite ground set UU such that every maximal stable set of the graph corresponds to a partition of UU. In 20092009, Orlin proved that every general partition graph is equistable, and conjectured that the converse holds as well. Orlin's conjecture, if true, would imply the conjecture due to Mahadev, Peled, and Sun. An intermediate conjecture, one that would follow from Orlin's conjecture and would imply the conjecture by Mahadev, Peled, and Sun, was posed by Miklavi\v{c} and Milani\v{c} in 20112011, and states that every equistable graph has a clique intersecting all maximal stable sets. The above conjectures have been verified for several graph classes. We introduce the notion of equistarable graphs and based on it construct counterexamples to all three conjectures within the class of complements of line graphs of triangle-free graphs
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