We force from large cardinals a model of ZFC in which ${\aleph }_{\omega +1}$ and ${\aleph }_{\omega +2}$ both have the tree property. We also prove that if we strengthen the large cardinal assumptions, then in the final model ${\aleph }_{\omega +2}$ even satisfies the super tree property.

Let $L(n,d)$ denote the minimum possible number of leaves in a tree of order $n$ and diameter $d.$ Lesniak (1975) gave the lower bound $B(n,d)=\lceil 2(n-1)/d\rceil$ for $L(n,d).$ When $d$ is even, $B(n,d)=L(n,d).$ But when $d$ is odd, $B(n,d)$ is smaller than $L(n,d)$ in general. For example, $B(21,3)=14$ while $L(21,3)=19.$ In this note, we determine $L(n,d)$ using new ideas. We also consider the converse problem and determine the minimum possible diameter of a tree with given order and number of leaves.

A graph $G$ is a $k$-tree if either $G$ is the complete graph on $k+1$ vertices, or $G$ has a vertex $v$ whose neighborhood is a clique of order $k$ and the graph obtained by removing $v$ from $G$ is also a $k$-tree. Clearly, a $k$-tree has at least $k+1$ vertices, and $G$ is a 1-tree (usual tree) if and only if it is a $1$-connected graph and has no $K_3$-minor. In this paper, motivated by some properties of 2-trees, we obtain a characterization of $k$-trees as follows: if $G$ is a graph with at least $k+1$ vertices, then $G$ is...

For a connected graph $G=(V,E)$ and a set $S\subseteq V\left(G\right)$ with at least two vertices, an $S$-Steiner tree is a subgraph $T=(V^{\text{'}},E^{\text{'}})$ of $G$ that is a tree with $S\subseteq V^{\text{'}}$. If the degree of each vertex of $S$ in $T$ is equal to 1, then $T$ is called a pendant $S$-Steiner tree. Two $S$-Steiner trees are if they share no vertices other than $S$ and have no edges in common. For $S\subseteq V\left(G\right)$ and $\left|S\right|\ge 2$, the pendant tree-connectivity ${\tau }_G\left(S\right)$ is the maximum number of internally disjoint pendant $S$-Steiner trees in $G$, and for $k\ge 2$, the $k$-pendant tree-connectivity ${\tau }_k\left(G\right)$ is the...

We prove that ${♦}^{*}$ implies there is a zero-dimensional Hausdorff Lindelöf space of cardinality $2^{{\aleph }_1}$ which has points $G_{\delta }$. In addition, this space has the property that it need not be Lindelöf after countably closed forcing.

Let $T$ be a tree. Then a vertex of $T$ with degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$. The set of leaves of $T$ is denoted by $L\left(T\right)$ and the set of branch vertices of $T$ is denoted by $B\left(T\right)$. For two distinct vertices $u$, $v$ of $T$, let $P_T[u,v]$ denote the unique path in $T$ connecting $u$ and $v.$ Let $T$ be a tree with $B\left(T\right)\ne \varnothing$. For each leaf $x$ of $T$, let $y_x$ denote the nearest branch vertex to $x$. We delete $V\left(P_T[x,y_x]\right)\setminus \left\{y_x\right\}$ from $T$ for all $x\in L\left(T\right)$. The resulting subtree of $T$ is called the reducible stem...

Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put ${}_{\lambda }\left(\alpha \right)=s\in \left(\alpha \right):s\left(0\right)=\lambda =min\left[s\right(\beta ):\beta <\alpha ]$. We show that f ∈ (α) iff for some natural number n there are infinite cardinals $\lambda ₀i>\lambda ₁>...>{\lambda }_{n-1}$ and ordinals $\alpha ₀,...,{\alpha }_{n-1}$ such that $\alpha =\alpha ₀+\cdots +{\alpha }_{n-1}$ and $f=f₀⏜f₁⏜...⏜f_{n-1}$ where each $f_i{\in }_{{\lambda }_i}\left({\alpha }_i\right)$. Under GCH we prove that if α < ω₂ then (i) ${}_{\omega }\left(\alpha \right)=s{\in }^{\alpha }\omega ,\omega ₁:s\left(0\right)=\omega$; (ii) if λ > cf(λ) = ω, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega ₁-closedin\alpha$; (iii) if cf(λ) = ω₁, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega -closedandsuccessor-closedin\alpha$; (iv) if cf(λ) > ω₁, ${}_{\lambda }\left(\alpha \right){=}^{\alpha }\lambda$. This yields a complete characterization of the classes (α) for all...

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...

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopěnka’s principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class $𝒮$ of morphisms in a locally presentable category $𝒞$ of structures, the orthogonal class of objects...

We prove that if $X$ is a first countable space with property $\left(DC\left({\omega }_1\right)\right)$ and with a $G_{\delta }$-diagonal then the cardinality of $X$ is at most $𝔠$. We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $𝔠$.

We show that ${\omega }^{*}\setminus \left\{p\right\}$ is not normal, if $p$ is a limit point of some countable subset of ${\omega }^{*}$, consisting of points of character ${\omega }_1$. Moreover, such a point $p$ is a Kunen point and a super Kunen point.

The Sombor index $SO\left(G\right)$ of a graph $G$ is the sum of the edge weights $\sqrt{d_G^2\left(u\right)+d_G^2\left(v\right)}$ of all edges $uv$ of $G$, where $d_G\left(u\right)$ denotes the degree of the vertex $u$ in $G$. A connected graph $G=(V,E)$ is called a quasi-tree if there exists $u\in V\left(G\right)$ such that $G-u$ is a tree. Denote $𝒬(n,k)=\{G:G$ is a quasi-tree graph of order $n$ with $G-u$ being a tree and $d_G\left(u\right)=k\}$. We determined the minimum and the second minimum Sombor indices of all quasi-trees in $𝒬(n,k)$. Furthermore, we characterized the corresponding extremal graphs, respectively.