We show that if a colouring c establishes ω₂ ↛ [(ω₁:ω)]² then c establishes this negative partition relation in each Cohen-generic extension of the ground model, i.e. this property of c is Cohen-indestructible. This result yields a negative answer to a question of Erdős and Hajnal: it is consistent that GCH holds and there is a colouring c:[ω₂]² → 2 establishing ω₂ ↛ [(ω₁:ω)]₂ such that some colouring g:[ω₁]² → 2 does not embed into c. It is also consistent that $2^{\omega ₁}$ is arbitrarily large,...

The space $\mathrm{\mathscr{H}𝒦}$ of Henstock-Kurzweil integrable functions on $[a,b]$ is the uncountable union of Fréchet spaces $\mathrm{\mathscr{H}𝒦}\left(X\right)$. In this paper, on each Fréchet space $\mathrm{\mathscr{H}𝒦}\left(X\right)$, an $F$-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space. It is known that every control-convergent sequence in the $\mathrm{\mathscr{H}𝒦}$ space always belongs to a $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space for some $X$. We illustrate how...

Let $p\in (0,1)$ be a real number and let $n\ge 2$ be an even integer. We determine the largest value $c_n\left(p\right)$ such that the inequality $$\sum _{i=1}^n{\left|a_i\right|}^p\ge c_n\left(p\right)$$ holds for all real numbers $a_1,...,a_n$ which are pairwise distinct and satisfy ${min}_{i\ne j}|a_i-a_j|=1$. Our theorem completes results of Ozeki, Mitrinović-Kalajdžić, and Russell, who found the optimal value $c_n\left(p\right)$ in the case $p>0$ and $n$ odd, and in the case $p\ge 1$ and $n$ even.

The Katětov ordering of two maximal almost disjoint (MAD) families $𝒜$ and $ℬ$ is defined as follows: We say that $𝒜{\le }_Kℬ$ if there is a function $f:\omega \to \omega$ such that $f^{-1}\left(A\right)\in ℐ\left(ℬ\right)$ for every $A\in ℐ\left(𝒜\right)$. In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called $K$-uniform if for every $X\in ℐ{\left(𝒜\right)}^{+}$, we have that ${𝒜|}_X{\le }_K𝒜$. We prove that CH implies that for every $K$-uniform MAD family $𝒜$ there is a $P$-point $p$ of ${\omega }^{*}$ such that the set of all Rudin-Keisler predecessors of $p$ is dense...

Let $P$ be a discrete multidimensional probability distribution over a finite set of variables $N$ which is only partially specified by the requirement that it has prescribed given marginals $\{P_A;A\in 𝒮\}$, where $𝒮$ is a class of subsets of $N$ with $\bigcup 𝒮=N$. The paper deals with the problem of approximating $P$ on the basis of those given marginals. The divergence of an approximation $\widehat{P}$ from $P$ is measured by the relative entropy $H\left(P\right|\widehat{P})$. Two methods for approximating $P$ are compared. One of them uses formerly introduced...

Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of $L_{\lambda ⁺\omega }$, with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with $⟨D^{ℳ},{\prec }^{ℳ}⟩$ a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where $⟨D^{{ℳ}^{\text{'}}},{\prec }^{{ℳ}^{\text{'}}}⟩$ is non-well-ordered. One of the interesting properties of this number is that the Hanf number of $L_{\lambda ⁺\omega }$ is exactly ${\beth }_{\delta \left(\lambda \right)}$. It was proved in [BK71] that if ℵ₀ < λ < κ$areregularcardinalnumbers,thenthereisaforcingextension,preservingcofinalities,suchthatintheextension$2λ = κ$and\delta \left(\lambda \right)<\lambda ⁺⁺.Weimprovethisresultbyprovingthefollowing:Suppose\aleph ₀<\lambda <\theta \le \kappa arecardinalnumberssuchthat$∙ ${\lambda }^{<\lambda }=\lambda$; ∙ cf(θ) ≥ λ⁺ and ${\mu }^{\lambda }<\theta$ whenever μ < θ; ∙ ${\kappa }^{\lambda }=\kappa$. Then there...

CONTENTSIntroduction......................................................................................................................................... 51. n-Monotonic functions on (— ∞, ∞)........................................................................................... 62. Order relations in the set of probability distribution functions....................................................... 12 2.1. Preliminary concepts...............................................................................................................

The well-known Impossibility Theorem of Arrow asserts that any generalized social welfare function (GSWF) with at least three alternatives, which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a dictatorship, is necessarily non-transitive. In 2002, Kalai asked whether one can obtain the following quantitative version of the theorem: For any $ϵ>0$, there exists $\delta =\delta \left(ϵ\right)$ such that if a GSWF on three alternatives satisfies the IIA condition and its probability of...

A proper coloring $c:V\left(G\right)\to \{1,2,...,k\}$, $k\ge 2$ of a graph $G$ is called a graceful $k$-coloring if the induced edge coloring $c^{\text{'}}:E\left(G\right)\to \{1,2,...,k-1\}$ defined by $c^{\text{'}}\left(uv\right)=|c\left(u\right)-c\left(v\right)|$ for each edge $uv$ of $G$ is also proper. The minimum integer $k$ for which $G$ has a graceful $k$-coloring is the graceful chromatic number ${\chi }_g\left(G\right)$. It is known that if $T$ is a tree with maximum degree $\Delta$, then ${\chi }_g\left(T\right)\le \lceil \frac{5}{3}\Delta \rceil$ and this bound is best possible. It is shown for each integer $\Delta \ge 2$ that there is an infinite class of trees $T$ with maximum degree $\Delta$ such that ${\chi }_g\left(T\right)=\lceil \frac{5}{3}\Delta \rceil$. In particular, we investigate for each...

Inspired by a result from Leibov, we find that the supremum defining the $BLO$ norm in $[0,1]$ is actually attained by a specific sub-interval of $[0,1]$ for $f\in VLO([0,1])$