Masasi Higasikawa
We consider embeddings between infinite graphs. In particular, We establish that there is no universal element in the class of countable graphs into which the random graph is not embeddable.
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A short proof of the existence of a minor-universal countable planar graph
本文提出了一种新的、更简短的次泛通用平面图构造方法，证明了其存在性。
Huynh et al. recently showed that a countable graph $G$ which contains every countable planar graph as a subgraph must contain arbitrarily large finite complete graphs as topological minors, and an infinite complete graph as a minor. We strengthen this result by showing that the same conclusion holds, if $G$ contains every countable planar graph as a topological minor. In particular, there is no countable planar graph containing every countable planar graph as a topological minor, answering a question by Diestel and Kühn. Moreover, we construct a locally finite planar graph which contains ever…
Universality in minor-closed graph classes
研究极小闭图类中普适图的存在性，通过构造具有线性着色数、线性扩张和平衡分离器的可数图，证明了包含所有可数平面图的图必须包含无限完全图作为子式，并给出了最优性结果。
Universal families of rayless graphs
研究无射线图的通用族存在性与复杂度，证明无限基数下存在强通用族，并探讨禁止子图对复杂度的影响。
Let C be a finite connected graph for which there is a countable universal C-free graph, and whose tree of blocks is a path. Then the blocks of C are complete. This generalizes a result of Furedi and Komjath, and fits naturally into a set of conjectures regarding the existence of countable C-free graphs, with C an arbitrary finite connected graph.
We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $Γ$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $Γ$. Further structural information about End $R$ is established including that Aut $Γ$ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.
We study the automorphism groups of countable homogeneous directed graphs (and some additional homogeneous structures) from the point of view of topological dynamics. We determine precisely which of these automorphism groups are amenable (in their natural topologies). For those which are amenable, we determine whether they are uniquely ergodic, leaving unsettled precisely one case (the "semi-generic" complete multipartite directed graph). We also consider the Hrushovski property. For most of our results we use the various techniques of [3], suitably generalized to a context in which the univer…
We give new examples and describe the complete lists of all measures on the set of countable homogeneous universal graphs and $K_s$-free homogeneous universal graphs (for $s\geq 3$) that are invariant with respect to the group of all permutations of the vertices. Such measures can be regarded as random graphs (respectively, random $K_s$-free graphs). The well-known example of Erdös--Rényi (ER) of the random graph corresponds to the Bernoulli measure on the set of adjacency matrices. For the case of the universal $K_s$-free graphs there were no previously known examples of the invariant measure…
We relate the existence problem of universal objects to the properties of corresponding enriched categories (lifts or expansions). In particular, extending earlier results, we prove that for every (possibly infinite) regular set F of finite connected structures there exists a (countable) ω-categorical universal structure U for the class Forb(F) (of all countable structures not containing any homomorphic image of a member of F). We employ a technique known as homogenization: The universal object U is the shadow (reduct) of an ultrahomogeneous structure U'. We also put the results of this paper …
Universal graphs for the topological minor relation
本文证明可数平面图类中不存在拓扑子式万有图，回答了Diestel和Kühn的问题并加强了Pach的结果，同时刻画了细分星图T使得可数T-自由图类具有拓扑子式万有图的条件。
We show that the existence of a universal structure implies the existence of a generic structure for any approximable class $\mathcal{C}$ of countable structures. We also show that the converse is not true. As a consequence, we provide several new examples of weak Fraïssé classes of finite graphs. Finally, we show that the class of all countable $C_4$-free graphs does not contain a generic structure, strengthening a result of A. Hajnal and J. Pach.
We consider a random geometric graph model, where pairs of vertices are points in a metric space and edges are formed independently with fixed probability $p$ between pairs within threshold distance $δ$. A countable dense set in a metric space is {\sl Rado} if this random model gives, with probability 1, a graph that is unique up to isomorphism. In earlier work, the first two authors proved that in finite dimensional spaces $\mathbb{R}^n$ equipped with the $\ell_{\infty}$ norm, all countable dense set satisfying a mild non-integrality condition are Rado. In this paper, we extend this result to…
We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite presentation). Extending classical work of Rado (for the random graph), we find a finite presentation for each of the following classes: homogeneous undirected graphs, homogeneous tournaments and homogeneous partially ordered sets. We also give a finite presentation of the rational Urysohn metric space and some homogeneous directed graphs. We survey well know…
Countable Graphs with Finite Path-width: Characterisation and Universality
该文刻画了有限路宽可数图的特征，并证明线宽至多为$k$的图类存在线宽为$\boldsymbol{O(k^2)}$的普适图，同时给出有限路宽图的普适性相关结论。
Universal graphs with forbidden wheel minors
本文证明对任意禁止轮图子式的可数图类，存在一个泛图包含该类中所有图作为诱导子图。
A set of graphs is said to be independent if there is no homomorphism between distinct graphs from the set. We consider the existence problems related to the independent sets of countable graphs. While the maximal size of an independent set of countable graphs is 2^omega the On Line problem of extending an independent set to a larger independent set is much harder. We prove here that singletons can be extended (``partnership theorem''). While this is the best possible in general, we give structural conditions which guarantee independent extensions of larger independent sets. This is related to…
Given a countable graph, we say a set $A$ of its vertices is \emph{universal} if it contains every countable graph as an induced subgraph, and $A$ is \emph{weakly universal} if it contains every finite graph as an induced subgraph. We show that, for almost every graph on $\mathbb N$, $(1)$ every set of positive upper density is universal, and $(2)$ every set with divergent reciprocal sums is weakly universal. We show that the second result is sharp (i.e., a random graph on $\mathbb N$ will almost surely contain non-universal sets with divergent reciprocal sums) and, more generally, that neithe…
We prove the existence of Rado sets in the Banach space of continuous functions on [0,1]. A countable dense set S is Rado if with probability 1, the infinite geometric random graph on S, formed by probabilistically making adjacent elements of S that are within unit distance of each other, is unique up to isomorphism. We show that for a suitable measure which we construct, almost all countable dense sets in the subspaces of piecewise linear functions and of polynomials are Rado. Moreover, all graphs arising from such sets are of a unique isomorphism type. For the subspace of Brownian motion pat…
The independence density of a finite hypergraph is the probability that a subset of vertices, chosen uniformly at random contains no hyperedges. Independence densities can be generalized to countable hypergraphs using limits. We show that, in fact, every positive independence density of a countably infinite hypergraph with hyperedges of bounded size is equal to the independence density of some finite hypergraph whose hyperedges are no larger than those in the infinite hypergraph. This answers a question of Bonato, Brown, Kemkes, and Prałat about independence densities of graphs. Furthermore, w…
A graph $U$ is universal for a graph class $\mathcal{C}\ni U$, if every $G\in \mathcal{C}$ is a minor of $U$. We prove the existence or absence of universal graphs in several natural graph classes, including graphs component-wise embeddable into a surface, and graphs forbidding $K_5$, or $K_{3,3}$, or $K_\infty$ as a minor. We prove the existence of uncountably many minor-closed classes of countable graphs that (do and) do not have a universal element. Some of our results and questions may be of interest to the finite graph theorist. In particular, one of our side-results is that every $K_5$-m…
We show that the problem of the existence of universal graphs with specified forbidden subgraphs can be systematically reduced to certain critical cases by a simple pruning technique which simplifies the underlying structure of the forbidden graphs, viewed as trees of blocks. As an application, we characterize the trees T for which a universal countable T-free graph exists.
The distinguishing number of a graph $G$ is the smallest positive integer $r$ such that $G$ has a labeling of its vertices with $r$ labels for which there is no non-trivial automorphism of $G$ preserving these labels. Albertson and Collins computed the distinguishing number for various finite graphs, and Imrich, Klavžar and Trofimov computed the distinguishing number of some infinite graphs, showing in particular that the Random Graph has distinguishing number 2. We compute the distinguishing number of various other finite and countable homogeneous structures, including undirected and directed…
The random ordered graph is the up to isomorphism unique countable homogeneous linearly ordered graph that embeds all finite linearly ordered graphs. We determine the reducts of the random ordered graph up to first-order interdefinability.
Bowtie-free graphs and generic automorphisms
本文证明了可数泛型omega范畴无蝶形图允许泛型自同构，并指出该图不可有限齐化。
Every countable graph can be built from finite graphs by a suitable infinite process, either adding new vertices randomly or imposing some rules on the new edges. On the other hand, a profinite topological graph is built as the inverse limit of finite graphs with graph epimorphisms. We propose to look at both constructions simultaneously. We consider countable graphs that can be built from finite ones by using both embeddings and projections, possibly adding a single vertex at each step. We show that the Rado graph can be built this way, while Henson's universal triangle-free graph cannot. We …
Faithful universal graphs for minor-closed classes
研究minor-闭类的忠实通用图，证明K_t-minor-无且含所有n顶点平面图的图需2^{Ω(n)}顶点，构造多项式大小的K_4-minor-无图（含所有n顶点树）和K_7-minor-无图（含所有n顶点K_4-minor-无图），回答相关问题并推广研究各类图的通用图阶。
The celebrated Erdős-Hajnal Conjecture says that in any proper hereditary class of finite graphs we are guaranteed to have a clique or anti-clique of size $n^c$, which is a much better bound than the logarithmic size that is provided by Ramsey's Theorem in general. On the other hand, in uncountable cardinalities, the model-theoretic property of stability guarantees a uniform set much larger than the bound provided by the Erdős-Rado Theorem in general. Even though the consequences of stability in the finite have been much studied in the literature, the countable setting seems a priori quite dif…
We prove local convergence results for the uniformly random, labelled or unlabelled, graphs from subcritical families. As an example special case, we prove Benjamini-Schramm convergence for the uniform random unlabelled tree. We introduce a compactification of the space of countable (connected) rooted graphs, and use it to generalise the notion of Benjamini-Schramm convergence in order to allow for vertices of infinite degree in the limit object.
Random Geometric Graphs in Reflexive Banach Spaces
本文研究赋范空间中随机几何图的 Rado 性，通过刻画稠密子集上阶梯等距映射何时必为等距映射，证明严格凸自反空间中几乎所有可数稠密集强非 Rado，但 ℓ₂ 中仍存在 Rado 集，并构造所有稠密集均强非 Rado 的 Banach 空间。
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