Shelah’s revised GCH theorem and a question by Alon on infinite graphs colorings

For every graph G, the coloring number of G does not exceed the least strong limit cardinal above the graph’s list-chromatic number.     

Generalizing Tutte’s theorem and maximal non-matchable graphs

A graph $G$ has a perfect matching if and only if $0$ is not a root of its matching polynomial $\mu (G,x)$. Thus, Tutte’s famous theorem asserts that $0$ is not a root of $\mu (G,x)$ if and only if $c_{\mathrm{odd}}(G?S)\le \left|S\right|$ for all $S?V\left(G\right)$, where $c_{\mathrm{odd}}\left(G\right)$ denotes the number of odd components of $G$. Tutte’s theorem can be proved using a characterization of the structure of maximal non-matchable graphs, that is, the edge-maximal graphs among those having no perfect matching. In this paper, we prove a generalized version of Tutte’s theorem in terms of avoiding any given real number $\theta$ as a root of $\mu (G,x)$. We also extend maximal non-matchable graphs to maximal $\theta$-non-matchable graphs and determine the structure of such graphs.     

Graph-like continua, augmenting arcs, and Menger’s theorem

We show that an adaptation of the augmenting path method for graphs proves Menger’s Theorem for wide classes of topological spaces. For example, it holds for locally compact, locally connected, metric spaces, as already known. The method lends itself particularly well to another class of spaces, namely the locally arcwise connected, hereditarily locally connected, metric spaces. Finally, it applies to every space where every point can be separated from every closed set not containing it by a finite set, in particular to every subspace of the Freudenthal compactification of a locally finite, connected graph. While closed subsets of such a space behave nicely in that they are compact and locally connected (and therefore locally arcwise connected), the general subspaces do not: They may be connected without being arcwise connected. Nevertheless, they satisfy Menger’s Theorem. This work was carried out while Antoine Vella was a Marie Curie Fellow at the Technical University of Denmark, as part of the research project TOPGRAPHS (Contract MEIF-CT-2005-009922), under the supervision of Carsten Thomassen.     

Kőnig’s edge-colouring theorem for all graphs

We show that the maximum degree of a graph G$G$ is equal to the minimum number of ocm sets covering  G$G$, where an ocm set is the vertex-disjoint union of elementary odd cycles and one matching, and a collection of ocm sets covers   G$G$ if every edge is in the matching of an ocm set or in some odd cycle of at least two ocm sets.     

Hall’s and Kőnig’s theorem in graphs and hypergraphs

We investigate the relation between Hall’s theorem and K?nig’s theorem in graphs and hypergraphs. In particular, we characterize the graphs satisfying a deficiency version of Hall’s theorem, thereby showing that this class strictly contains all K?nig–Egerváry graphs. Furthermore, we give a generalization of Hall’s theorem to normal hypergraphs.     

Hadwiger’s conjecture for uncountable graphs

We consider the infinite form of Hadwiger’s conjecture. We give a(n apparently novel) proof of Halin’s 1967 theorem stating that every graph X with coloring number \(>\kappa \) (specifically with chromatic number \(>\kappa \)) contains a subdivision of \(K_\kappa \). We also prove that there is a graph of cardinality \(2^\kappa \) and chromatic number \(\kappa ^+\) which does not contain \(K_{\kappa ^+}\) as a minor. Further, it is consistent that every graph of size and chromatic number \(\aleph _1\) contains a subdivision of \(K_{\aleph _1}\).     

Blackwell’s theorem for fuzzy variables

Recently, Zhao et al. (Euro J Oper Res 169:189–201, 2006) discussed a random fuzzy renewal process based on the random fuzzy theory and established Blackwell’s theorem in random fuzzy sense. They obtained Blackwell’s theorem for fuzzy variables by degenerating the process. However, this result is invalid. We provide some counterexamples and offer a corrected version of fuzzy Blackwell’s theorem.     

Moufang’s theorem for non-Moufang loops

We introduce a class of non-Moufang loops satisfying Moufang’s theorem.     

Engel’s theorem for malcev superalgebras

A version of Engel’s theorem for Malcev superalgebras is proved in the spirit of theJacobson-Engel theorem for Lie algebras. Some consequences for the structure of Malcev superalgebras with trivial Lie nucleus are derived.     

Siegel’s theorem for Drinfeld modules

We prove a Siegel type statement for finitely generated -submodules of under the action of a Drinfeld module . This provides a positive answer to a question we asked in a previous paper. We also prove an analog for Drinfeld modules of a theorem of Silverman for nonconstant rational maps of over a number field.     

Ando’s theorem for nonnegative forms

In this paper, we present a generalization of Ando??s theorem for nonnegative forms. He proved that the infimum of two positive operators A and B exists in the positive cone if and only if the generalized shorts [B]A and [A]B are comparable (see Ando et?al. in Problem of infimum in the positive cone, analytic and geometric inequalities and applications, Math. Appl. 478, pp 1?C12, 1999). That is, [A]B??? [B]A or [B]A??? [A]B. Using the concept of the parallel sum of nonnegative forms, Hassi, Sebestyén and de Snoo investigated the decomposability of a nonnegative form ${\mathfrak{t}}$ into an almost dominated and a singular part with respect to a nonnegative form ${\mathfrak{w}}$ (see Hassi et?al. in J. Funct. Anal. 257(12), 3858?C3894, 2009). Applying their results, we formulate a necessary and sufficient condition for the existence of the infimum of two nonnegative forms.     

Alperin’s fusion theorem for localities

In these notes we give a version of the Alperin–Goldschmidt fusion theorem for localities.     

Aubin’s Lemma for the Yamabe constants of infinite coverings and a positive mass theorem

Aubin’s Lemma says that, if the Yamabe constant of a closed conformal manifold (M, C) is positive, then it is strictly less than the Yamabe constant of any of its non-trivial finite conformal coverings. We generalize this lemma to the one for the Yamabe constant of any (M _∞, C _∞) of its infinite conformal coverings, provided that π ₁(M) has a descending chain of finite index subgroups tending to π ₁(M _∞). Moreover, if the covering M _∞ is normal, the limit of the Yamabe constants of the finite conformal coverings (associated to the descending chain) is equal to that of (M _∞, C _∞). For the proof of this, we also establish a version of positive mass theorem for a specific class of asymptotically flat manifolds with singularities.     

Naishul’s theorem for fibered holomorphic maps

We show that the fibered rotation number associated to an indifferent invariant curve for a fibered holomorphic map is a topological invariant.     

Beurling’s analyticity theorem for quantum differences

A theorem of Beurling states that if f satisfies , n = 1, 2,..., for some 0 < ρ < 2, on a real interval I, then f is analytic in a rhombus containing I. We study the corresponding problem for the quantum differences Δ _n f (q, x), q > 1, n = 1, 2,..., for functions defined on (0, ∞) and prove quantitative and qualitative analogues of Beurling’s result. We also characterize the analyticity of f on subintervals of (0, ∞) in q-analytic terms.     

Dugundji’s theorem for cone metric spaces

In this work, we will prove the Dugundji extension theorem for the cone metric space. It is heavily reliant on the paracompactness of the cone topology that is proved by Ayse Sönmez in the paper Sönmez (2010) [11].     

Hechler’s theorem for the null ideal

We prove the following theorem: For a partially ordered set Q such that every countable subset of Q has a strict upper bound, there is a forcing notion satisfying the countable chain condition such that, in the forcing extension, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechlers classical result in the theory of forcing. The corresponding theorem for the meager ideal was established by Bartoszyski and Kada.Research supported by NSERC. The first author thanks F.D. Tall and the Department of Mathematics at the University of Toronto for their hospitality during the academic year 2003/2004 when the present paper was completed.The second author was supported by Grant-in-Aid for Young Scientists (B) 14740058, MEXT.Mathematics Subject Classification (2000): 03E35, 03E17Revised version: 16 February 2004     

Lazard’s theorem for S-pure flat modules

Let R be an associative ring with identity. For a given class 𝒮 of finitely presented left (respectively right) R-modules containing R, we present a complete characterization of 𝒮-pure injective modules and 𝒮-pure flat modules. Consider that 𝒮 is a class of (R,R)-bimodules containing R with the following property: every element of 𝒮 is a finitely presented left and right R-module. We give a necessary and sufficient condition for 𝒮 to have Lazard’s theorem, and then we present our desired Lazard’s theorem.