Abstract Recent work showing the existence of conflict-free almost-perfect hypergraph matchings has found many applications. We show that, assuming certain simple degree and codegree conditions on the hypergraph $ \mathcal{H}$ and the conflicts to be avoided, a conflict-free almost-perfect matching can be extended to one covering all vertices in a particular subset of $ V(\mathcal{H})$ , by using an additional set of edges; in particular, we ensure that our matching avoids all additional conflicts, which may consist of both old and new edges. This setup is useful for various applications in design theory and Ramsey theory. For example, our main result provides a crucial tool in the recent proof of the high-girth existence conjecture due to Delcourt and Postle. It also provides a black box which encapsulates many long and tedious calculations, greatly simplifying the proofs of results in generalised Ramsey theory.

Abstract The thesis is divided into two parts. The first one focuses on generalized descriptive set theory, and the second one on combinatorics, model theory, and Ramsey theory. Generalized descriptive set theory (GDST) is a natural extension of (classical) descriptive set theory (DST) where countable is replaced by uncountable. But the framework of GDST is narrow if compared to that of DST, as so far GDST has mostly concentrated on the study of the generalized Baire space , rather than considering arbitrary “Polish-like” spaces or standard $\kappa $ -Borel spaces. Also, GDST is usually developed for cardinals satisfying $\kappa ^{<\kappa }=\kappa $ , which implies that $\kappa $ must be regular. The goal of the first part of the thesis is to fill these gaps, studying classes of spaces that could take the role of Polish spaces in the generalized context under the weak assumption $2^{<\kappa }=\kappa $ , which allows one to include singular cardinals and can consistently hold at every cardinal (e.g., in models of $ \mathsf {ZFC+GCH} $ ). In Chapter 1, we begin by considering the case when $\kappa $ is regular. We consider several candidates for “Polish-like” spaces that have been proposed in the literature (e.g., $\mathbb {G}$ -Polish spaces and $\mathrm {SC}_\kappa $ -spaces), and introduce a new one ( $f\mathrm {SC}_\kappa $ -spaces). We show that all these classes are nicely organized in four groups, with two clear dividing lines between them: $ \kappa $ -additivity, which can be interpreted as a strong analog of zero-dimensionality, and the degree of completeness (Figure 1). Figure 1 Relationships among Polish-like classes of regular Hausdorff spaces of weight $\leq \kappa $ , for a totally ordered Abelian group $\mathbb {G}$ of degree $\deg (\mathbb {G})=\kappa>\omega $ . All the proposed classes give rise to the same class of spaces up to $ \kappa $ -Borel isomorphism, providing a natural setup to work with. Then, various results from classical DST about Polish, Borel, and standard Borel spaces are extended to this context. Chapter 2 extends the previous analysis to embrace singular cardinals too. In particular, it contains an in-depth study of the generalizations and characterizations of metrizability necessary in the singular case. The main result on this topic is a new metrization theorem in terms of topological games that holds for both classical metrizability and $\mathbb {G}$ -metrizability. Chapter 3 features various examples of spaces in the classes considered above, and a study of linearly ordered topological spaces (LOTS) and generalized ordered spaces (GO-spaces) in the context of GDST. The second part of the thesis deals with a recently discovered notion in combinatorics. In 2019, Solecki introduced the classes of Ramsey monoids and $\mathbb {Y}$ -controllable monoids to collect and extend different theorems in combinatorics, like Hindman’s Finite Sum Theorem, Carlson’s Theorem, Gowers’ FIN $_\kappa $ Theorem, and Furstenberg–Katznelson’s Ramsey Theorem. Then, he provided a necessary condition and some sufficient conditions for a finite monoid to be Ramsey or $\mathbb {Y}$ -controllable. Chapters 4 and 5 aim to continue the work started by Solecki on these and other related classes of monoids. We improve the necessary conditions and the sufficient conditions provided by Solecki, reaching in particular a full characterization of Ramsey monoids. This further extends results like Carlson’s Theorem and Gowers’ FIN $_\kappa $ Theorem, but it also sets a precise limit on when it is possible to obtain similar statements. We also give examples of classes of $\mathbb {Y}$ -controllable monoids that do not satisfy some of the sufficient conditions, suggesting possible strategies to improve the results we provided. Then, we show that in certain particular classes of $\mathbb {Y}$ -controllable monoids with stronger properties, the remaining sufficient conditions become necessary as well. In Chapter 5, we also study local versions of the classes of Ramsey and $\mathbb {Y}$ -controllable monoids that are better suited for infinite monoids. The thesis contains material from joint works with Luca Motto Ros, Philipp Schlicht, and Eugenio Colla. Abstract prepared by Claudio Agostini E-mail: agostini.claudio@renyi.hu Current affiliation: HUN-REN ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS REÁLTANODA UTCA 13-15 H-1053, BUDAPEST

Abstract We show that for every non-spherical set X in $\mathbb {E}^d$ , there exists a natural number m and a red/blue-coloring of $\mathbb {E}^n$ for every n such that there is no red copy of X and no blue progression of length m with each consecutive point at distance $1$ . This verifies a conjecture of Wu and the first author.

Ramsey theory studies how to find highly-ordered substructures within an otherwise unwieldy object. Ramsey theory is a highly active area of research in contemporary mathematics, with some mathematicians focusing on finite structures and others on infinite ones. In this survey paper, we will give an overview of a few topics in infinite Ramsey theory, with an emphasis on how set theory is involved. That is, we will focus on large, infinite objects and ask exactly how infinite they must be in order to ensure that we have infinite, highly-ordered substructures. After introducing the general idea in the finite case, we will prove Ramsey's theorem about infinite graphs. Then we will transition into questions about finding uncountably infinite, highly-ordered substructures. This will give us a convenient excuse to discuss infinities and independence results in set theory, as well as topological colorings. No knowledge of set theory or topology is required to understand this paper.

<p>In this paper, we consider Ramsey and Gallai-Ramsey numbers for a generalized fan <span class="math inline">\(F_{t,n}:=K_1+nK_t\)</span> versus triangles. Besides providing some general lower bounds, our main results include the evaluations of <span class="math inline">\(r(F_{3,2}, K_3)=13\)</span> and <span class="math inline">\(gr(F_{3,2}, K_3, K_3)=31\)</span>.</p>

Abstract A fundamental problem in Ramsey theory is to determine the growth rate in terms of $n$ of the Ramsey number $r(H, K_{n}^{(3)})$ of a fixed $3$-uniform hypergraph $H$ versus the complete $3$-uniform hypergraph with $n$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $H$, including links of odd cycles and tight cycles of length not divisible by three, $r(H, K_{n}^{(3)}) \ge 2^{\Omega _{H}(n \log n)}$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $H$ for which $r(H, K_{n}^{(3)})$ is superpolynomial in $n$. This provides the first example of a separation between $r(H,K_{n}^{(3)})$ and $r(H,K_{n,n,n}^{(3)})$, since the latter is known to be polynomial in $n$ when $H$ is linear.

Ramsey theory of the phase transitions of the second order

This paper explores various Ramsey numbers associated with cycles with pendant edges, including the classical Ramsey number, the star-critical Ramsey number, the Gallai–Ramsey number, and the star-critical Gallai–Ramsey number. These Ramsey numbers play a crucial role in combinatorial mathematics, determining the minimum number of vertices required to guarantee specific monochromatic substructures. We establish upper and lower bounds for each of these numbers, providing new insights into their behavior for cycles with pendant edges—graphs formed by attaching additional edges to one or more vertices of a cycle. The results presented contribute to the broader understanding of Ramsey theory and serve as a foundation for future research on generalized Ramsey numbers in complex graph structures.

Progressions in Euclidean Ramsey theory

Abstract Ramsey’s theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. In 1962, Erdős conjectured that the random 2-edge-coloring minimizes the number of monochromatic copies of $K_k$ , and the conjecture was extended by Burr and Rosta to all graphs. In the late 1980s, the conjectures were disproved by Thomason and Sidorenko, respectively. A classification of graphs whose number of monochromatic copies is minimized by the random 2-edge-coloring, which are referred to as common graphs, remains a challenging open problem. If Sidorenko’s conjecture, one of the most significant open problems in extremal graph theory, is true, then every 2-chromatic graph is common and, in fact, no 2-chromatic common graph unsettled for Sidorenko’s conjecture is known. While examples of 3-chromatic common graphs were known for a long time, the existence of a 4-chromatic common graph was open until 2012, and no common graph with a larger chromatic number is known. We construct connected k-chromatic common graphs for every k. This answers a question posed by Hatami et al. [Non-three-colourable common graphs exist, Combin. Probab. Comput. 21 (2012), 734–742], and a problem listed by Conlon et al. [Recent developments in graph Ramsey theory, in Surveys in combinatorics 2015, London Mathematical Society Lecture Note Series, vol. 424 (Cambridge University Press, Cambridge, 2015), 49–118, Problem 2.28]. This also answers in a stronger form the question raised by Jagger et al. [Multiplicities of subgraphs, Combinatorica 16 (1996), 123–131] whether there exists a common graph with chromatic number at least four.

Abstract We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs (i.e., $x,y\in {\mathbb N}$ such that $x^2\pm y^2=z^2$ for some $z\in {\mathbb N}$ ). We also show that partitions generated by level sets of multiplicative functions taking finitely many values always contain Pythagorean triples. Our proofs combine known Gowers uniformity properties of aperiodic multiplicative functions with a novel and rather flexible approach based on concentration estimates of multiplicative functions.

Application of Ramsey's growth theory to optimal taxation

Abstract A landmark result of Erdős, Ginzburg, and Ziv (EGZ) states that any sequence of $2n-1$ elements in ${\mathbb {Z}}/n$ contains a zero-sum subsequence of length n . While algebraic techniques have predominated in deriving many deep generalizations of this theorem over the past sixty years, here we introduce topological approaches to zero-sum problems which have proven fruitful in other combinatorial contexts. Our main result is a topological criterion for determining when any ${\mathbb {Z}}/n$ -coloring of an n -uniform hypergraph contains a zero-sum hyperedge. In addition to applications for Kneser hypergraphs, for complete hypergraphs our methods recover Olson’s generalization of the EGZ theorem for arbitrary finite groups. Furthermore, we give a fractional generalization of the EGZ theorem with applications to balanced set families and provide a constrained EGZ theorem which imposes combinatorial restrictions on zero-sum sequences in the original result.

Abstract Application of the Ramsey Infinite Theorem to the fundamental variational principles of physics is addressed. The Hamilton Least Action Principle states that, for a true/actual trajectory of a system, Hamilton’s Action is stationary for the paths, which evolve from the preset initial space-time point to the preset final space-time point. The Hamilton Principle distinguishes between the actual and trial/test trajectories of the system in the configurational space. This enables the transformation of the infinite set of points of the configurational space (available for the system) into the bi-colored, infinite, complete, graph, when the points of the configurational space are seen as the vertices, actual paths connecting the vertices/ points of the configurational space are colored with red; whereas, the trial links/paths are colored with green. Following the Ramsey Infinite Theorem, there exists the infinite, monochromatic sequence of the pathways/clique, which is completely made up of actual or virtual paths, linking the interim states of the system. The same procedure is applicable to the Maupertuis’s principle (classical and quantum), Hilbert-Einstein relativistic variational principle and reciprocal variational principles. Exemplifications of the Infinite Ramsey Theorem are addressed.

Abstract A graph is said to be Ramsey for a tuple of graphs if every ‐coloring of the edges of contains a monochromatic copy of in color , for some . A fundamental question at the intersection of Ramsey theory and the theory of random graphs is to determine the threshold at which the binomial random graph becomes asymptotically almost surely Ramsey for a fixed tuple , and a famous conjecture of Kohayakawa and Kreuter predicts this threshold. Earlier work of Mousset–Nenadov–Samotij, Bowtell–Hancock–Hyde, and Kuperwasser–Samotij–Wigderson has reduced this probabilistic problem to a deterministic graph decomposition conjecture. In this paper, we resolve this deterministic problem, thus proving the Kohayakawa–Kreuter conjecture. Along the way, we prove a number of novel graph decomposition results that may be of independent interest.

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$ , where the ai and b are fixed coefficients and h is an arbitrary integer polynomial of degree d. We establish that the natural necessary conditions for this equation to have a monochromatic non-constant solution with respect to any finite colouring of the prime numbers are also sufficient when the equation has at least $(1+o(1))d^2$ variables. We similarly characterize when such equations admit solutions over any set of primes with positive relative upper density. In both cases, we obtain lower bounds for the number of monochromatic or dense solutions in primes that are of the correct order of magnitude. Our main new ingredient is a uniform lower bound on the cardinality of a prime polynomial Bohr set.

We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. We also considered the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely (M1,g1) and (M2,g2), represented by the Riemann surfaces which intersect along the curve (M1,g1)∩(M2,g2)=ℒ were addressed. Curve ℒ does not contain geodesic lines in either of the manifolds (M1,g1) and (M2,g2). Consider six points located on the ℒ: {1,…6}⊂ℒ. The points {1,…6}⊂ℒ are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold (M1,g1)/red links, and, alternatively, with the geodesic lines belonging to the manifold (M2,g2)/green links. Points {1,…6}⊂ℒ form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented.

We classify the Boolean degree 1 1 functions of k k -spaces in a vector space of dimension n n (also known as Cameron-Liebler classes) over the field with q q elements for n ≥ n 0 ( k , q ) n \geq n_0(k, q) . This also implies that two-intersecting sets with respect to k k -spaces do not exist for n ≥ n 0 ( k , q ) n \geq n_0(k, q) . Our main ingredient is the Ramsey theory for geometric lattices.

A graph $G$ is a pairwise compatibility graph (PCG) if there exists an edge-weighted tree and an interval $I$, such that each leaf of the tree is a vertex of the graph, and there is an edge $\{ x, y \}$ in $G$ if and only if the weight of the path in the tree connecting $x$ and $y$ lies within the interval $I$. Originating in phylogenetics, PCGs are closely connected to important graph classes like leaf-powers and multi-threshold graphs, widely applied in bioinformatics, especially in understanding evolutionary processes. In this paper we introduce two natural generalizations of the PCG class, namely $k$-OR-PCG and $k$-AND-PCG, which are the classes of graphs that can be expressed as union and intersection, respectively, of $k$ PCGs. These classes can be also described using the concepts of the covering number and the intersection dimension of a graph in relation to the PCG class. We investigate how the classes of OR-PCG and AND-PCG are related to PCGs, $k$-interval-PCGs and other graph classes known in the literature. In particular, we provide upper bounds on the minimum $k$ for which an arbitrary graph $G$ belongs to $k$-interval-PCGs, $k$-OR-PCG or $k$-AND-PCG classes. For particular graph classes we improve these general bounds. Moreover, we show that, for every integer $k$, there exists a bipartite graph that is not in the $k$-interval-PCGs class, proving that there is no finite $k$ for which the $k$-interval-PCG class contains all the graphs. This answers an open question of Ahmed and Rahman from 2017. Finally, using a Ramsey theory argument, we show that for any $k$, there exists graphs that are not in $k$-AND-PCG, and graphs that are not in $k$-OR-PCG.

Word-representable graphs were originally introduced by Kitaev and Pyatkin, motivated by work of Kitaev and Seif in algebra. Since their introduction, however, there has been a great deal of work in understanding their graph theoretical properties. In this paper, we introduce tools from partially ordered sets, Ramsey theory as well as probabilistic methods to study them. Through these, we settle a number of open problems in the field, regarding both the existence and length of word-representations for various classes of graphs.