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In this paper, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht-Fraisse bijective pebble game in Hella's (Ann. Pure Appl. Log., 1989) hierarchy. This is a Spoiler-Duplicator game…
As is well known, a graph is a mathematical object modeling the existence of a certain relation between pairs of elements of a given set. Therefore, it is not surprising that many of the first results concerning graphs made reference to…
We introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for…
We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property. We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation…
In this paper, we give algorithms for determining the existence of isomorphism between two finite-dimensional Lie algebras and compute such an isomorphism in the affirrmative case. We also provide algorithms for determining algebraic…
An infinite graph is highly connected if the complement of any subgraph of smaller size is connected. We consider weaker versions of Ramsey's Theorem asserting that in any coloring of the edges of a complete graph there exist large highly…
Recent work by Atserias and Dawar (J. Log. Comp 2019) and Tucker-Foltz (LMCS 2024) has established undefinability results in fixed-point logic with counting (FPC) corresponding to many classical complexity results from the hardness of…
We answer a question posed by Hirschfeldt and Jockusch by showing that whenever $k > \ell$, Ramsey's theorem for singletons and $k$-colorings, $\mathsf{RT}^1_k$, is not strongly computably reducible to the stable Ramsey's theorem for…
Let kappa be an uncountable cardinal and the edges of a complete graph with kappa vertices be colored with aleph_0 colors. For kappa >2^{aleph_0} the Erd\H{o}s-Rado theorem implies that there is an infinite monochromatic subgraph. However,…
We provide a characterization of those relation algebras which are isomorphic to the algebras of compatible relations of some $\Z_2$-set. We further prove that this class is finitely axiomatizable in first-order logic in the language of…
We associate to each synchronous game an algebra whose representations determine if the game has a perfect deterministic strategy, perfect quantum strategy or one of several other perfect strategies. when applied to the graph coloring game,…
We show that approximate graph colouring is not solved by any level of the affine integer programming (AIP) hierarchy. To establish the result, we translate the problem of exhibiting a graph fooling a level of the AIP hierarchy into the…
Let $X$ be a (repetitive) infinite connected simple graph with a finite upper bound $\Delta$ on the vertex degrees. The main theorem states that $X$ admits a (repetitive) limit aperiodic vertex coloring by $\Delta$ colors. This refines a…
Two graphs are co-spectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are co-spectral with it are isomorphic to it. We consider these…
A $k$-colouring (not necessarily proper) of vertices of a graph is called {\it acyclic}, if for every pair of distinct colours $i$ and $j$ the subgraph induced by the edges whose endpoints have colours $i$ and $j$ is acyclic. In the paper…
We call a bipartite graph {\it homogeneous} if every finite partial automorphism which respects left and right can be extended to a total automorphism. A $(\kappa,{\lambda} )$ bipartite graph is a bipartite graph with left side of size…
Motivated by understanding non-strict and strict pure strategy equilibria in network anti-coordination games, we define notions of stable and, respectively, strictly stable colorings in graphs. We characterize the cases when such colorings…
Graph matching is one of the most important problems in graph theory and combinatorial optimization, with many applications in various domains. Although meta-heuristic algorithms have had good performance on many NP-Hard and NP-Complete…
A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree $\Delta$…
In this paper we shall give sufficient conditions for local CR diffeomorphisms between two real analytic submanifolds of $\Bbb C^N$ to be determined by finitely many derivatives at finitely many points. These conditions will also be shown…