We introduce a new method which combines Prikry forcing with an iteration between the Prikry points. Using our method we prove from large cardinals that it is consistent that the tree property holds at ℵₙ for n ≥ 2, ${\aleph }_{\omega }$ is strong limit and $2^{{\aleph }_{\omega }}={\aleph }_{\omega +2}$.

One way to generalize complete Erdős space ${}_c$ is to consider uncountable products of zero-dimensional $G_{\delta }$-subsets of the real line, intersected with an appropriate Banach space. The resulting (nonseparable) complete Erdős spaces can be fully classified by only two cardinal invariants, as done in an earlier paper of the authors together with J. van Mill. As we think this is the correct way to generalize the concept of complete Erdős space to a nonseparable setting, natural questions arise...

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...

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...

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.

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.

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...

The topology and combinatorial structure of the Mandelbrot set ${ℳ}^d$ (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in ${ℳ}^d$. Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, ${\Lambda }^d$. In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence...

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...

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...

Let T¹ₙ = (V,E₁) and T²ₙ = (V,E₂) be the trees on n vertices with $V=v₀,v₁,...,v_{n-1}$, $E₁=v₀v₁,...,v₀v_{n-3},v_{n-4}{v}_{n-2},v_{n-3}{v}_{n-1}$ and $E₂=v₀v₁,...,v₀v_{n-3},v_{n-3}{v}_{n-2},v_{n-3}{v}_{n-1}$. For p ≥ n ≥ 5 we obtain explicit formulas for ex(p;T¹ₙ) and ex(p;T²ₙ), where ex(p;L) denotes the maximal number of edges in a graph of order p not containing L as a subgraph. Let r(G₁,G₂) be the Ramsey number of the two graphs G₁ and G₂. We also obtain some explicit formulas for $r(Tₘ,T{ₙ}^i)$, where i ∈ 1,2 and Tₘ is a tree on m vertices with Δ(Tₘ) ≤ m - 3.

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...

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.

This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal κ is characterized by a Scott sentence ${\varphi }_{ℳ}$ if ${\varphi }_{ℳ}$ has a model of size κ, but no model of size κ⁺. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if ${\aleph }_{\beta }$ is characterized by a Scott sentence, then $2^{{\aleph }_{\beta +\beta ₁}}$ is (homogeneously) characterized by a Scott sentence, for all 0 <...

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...