We will survey some of the major directions of research in arithmetic combinatorics and their connections to other fields. We will then discuss three new results. The first result will generalize a structural theorem from Balog and Szemerédi. The second result will establish a new tool in incidence geometry, which should prove useful in attacking combinatorial estimates. The third result evolved from the famous sum-product problem, by providing a partial categorization of bivariate polynomial set functions which induce exponential expansion on all finite sets of real numbers.

La géométrie combinatoire est une large et belle branche des mathématiques. Cette thèse doctorale se compose de l'étude de cinq sujets différents dans ce domaine. Même si les problèmes et les techniques utilisés pour y faire face sont divers, ils partagent le mêeme objectif: Étudier l'interaction entre les structures combinatoires et géométriques. Dans le chapitre 1, nous étudions le problème suivant : pour un entier positif k, combien de points en position générale devons-nous prendre dans le plan de sorte que nous pouvons toujours trouver k d'entre eux définissant des triangles avec un rayon du cercle circonscrit distinct ? Cette question a été posée par Paul Erdös en 1975 qui a lui même proposé une solution en 1978. Toutefois, la preuve a omis par inadvertance un cas non trivial. Nous avons repris ce cas et donné une solution à la question en utilisant des outils de base de la géométrie algébrique et nous fournissons une borne polynomiale pour le nombre de points nécessaires.Dans le chapitre 2, nous sommes intéressés par de généralisations géométriques du critère de Hall pour les couplages dans les graphes bipartits (1935). Nous obtenons des théorèmes géométriques type Hall pour des ensembles convexes disjoints et pour points en position générale dans l'espace euclidien. Les outils de ce chapitre sont topologiques, et l'approche est motivés par une méthode remarquable introduite par Aharoni et Haxell en $2000$ ainsi que par ses généralisations.D'autre part, dans le chapitre 3, nous commençons par un théorème de Helly fractionné de 1979 due à A. Liu et M. Katchalski pour motiver un résultat combinatoire. Nous étudions des conditions combinatoires que des familles de graphes doivent avoir pour permettre d'obtenir des versions plus fine du théorème de Turán. Nous trouvons des liens intéressants entre les nombres de Turán, les nombres chromatiques et les nombres de clique dans la famille. Les outils de ce chapitre sont purement combinatoires.Dans le chapitre 4, nous nous concentrons sur l'obtention des résultats pour la bien connue classe des matroïde chemin du réseau introduite par Bonin, de Mier et Noy en 2003. La contribution principale est de prouver pour cette classe la validité d'une conjecture de Merino et Welsh (1999) sur une inégalité de certaines valeurs du polynôme de Tutte. Pour ce faire, nous introduisons et étudions des serpents, une classe spéciale de matroïdes chemin du réseau ``mince''.Enfin, dans le chapitre 5, nous étudions une variante d'un problème des transversales posé par J.L. Arocha, J. Bracho, L. Montejano et J.L. Ramírez-Alfonsín en 2010. Dans leur travaux originaux, ils ont rémarqué que si nous avons peu de points dans l'espace euclidien alors il est possible de trouver une transversale d'une dimension donnée qui travers les enveloppes convexes de tous les k-ensembles de points. De m&eme, ils montrent qu'il est impossible de trouver une telle transversale lorsque nous avons beaucoup de points. Les auteurs donnent des bornes spécifiques et ils laissent aussi quelques problèmes ouverts. Si la définition de transversale est légèrement plus restrictive, alors le problème peut être étudié en utilisant la théorie des matroïdes orientés. Dans la présente thèse, nous fournissons les détails de cette relation et nous donnons des bornes pour la famille de polytopes cycliques. / Combinatorial geometry is a broad and beautiful branch of mathematics. This PhD Thesis consists of the study of five different topics in this area. Even though the problems and the tools used to tackle them are diverse, they share a unifying goal: To explore the interaction between combinatorial and geometric structures.In Chapter 1 we study a problem by Paul Erdös: for a positive integer k, how many points in general position do we need in the plane so that we can always find a k-subset of them defining triangles with distinct circumradii? This question was posed in 1975 and Erdös himself proposed a solution in 1978. However, the proof inadvertently left out a non-trivial case. We deal with the case using basic tools from algebraic geometry and we provide a polynomial bound for the needed number of points.In Chapter 2 we are interested in providing geometric extensions of Hall's criterion for matchings in bipartite graphs (1935). We obtain geometric Hall-type theorems for pairwise disjoint convex sets and for points in general position in euclidean space. The tools of this chapter are topological, and are motivated by a remarkable method introduced by Aharoni and Haxell in 2000 and its generalizations.On the other hand, in Chapter 3 we begin with a fractional Helly theorem from 1979 by A. Liu and M. Katchalski to motivate a combinatorial result. We study combinatorial conditions on families of graphs that allow us to have sharpened variants of Turán's theorem. We find interesting relations between the Turán numbers, the chromatic numbers and the clique numbers of graphs in the family. The tools in this chapter are only combinatorial.In Chapter 4 we focus on obtaining some results for the well studied class of lattice path matroids introduced by Bonin, de Mier and Noy in 2003. The main contribution is proving for this class the validity of a 1999 conjecture of Merino and Welsh concerning an inequality involving certain values of the Tutte polynomial. In order to do this, we introduce and study snakes, a special class of ``thin'' lattice path matroids.Finally, in Chapter 5 we explore a variant of a transversal problem posed by J.L. Arocha, J. Bracho, L. Montejano and J.L. Ramírez-Alfonsín in 2010. In their original work, they realized that if we have few points in euclidean space then it is possible to find a transversal of a given dimension that goes through all the convex hulls of k-subsets of points. Similarly, they show that it is impossible to find such a transversal when we have many points. The authors give some specific bounds and they also leave some open problems. If the definition of transversal is slightly more restrictive, then the problem can be tackled using oriented matroid theory. We provide the details of the relation and we give bounds for the family of cyclic polytopes.

In this thesis we consider extensions and generalizations of some classical problems in Combinatorial Geometry. Our work is an offshoot of four classical problems in Combinatorial Geometry. A fundamental assumption in these problems is that the underlying point set is R2. Two fundamental themes entwining the problems considered in this thesis are: “What happens if we assume that the underlying point set is finite?”, “What happens if we assume that the underlying point set has a special structure?”. Let P ⊂ R2 be a finite set of points in general position. It is reasonable to expect that if |P| is large then certain ‘patterns’ in P always occur. One of the first results was the Erd˝os-Szekeres Theorem which showed that there exists a f(n) such that if |P| ≥ f(n) then there exists a convex subset S ⊆ P, |S| = n i.e., a subset which is a convex polygon of size n. A considerable number of such results have been found since. Avis et al. in 2001 posed the following question which we call the n-interior point problem: Is there a finite integer g(n) for every n, such that, every point set P with g(n) interior points has a convex subset S ⊆ P with n interior points. i.e. a subset which is a convex polygon that contains exactly n interior points. They showed that g(1) = 1, g(2) = 4. While it is known that g(3) = 9, it is not known whether g(n) exists for n ≥ 4. In the first part of this thesis, we give a positive solution to the n-interior point problem for point sets with bounded number of convex layers. We say a family of geometric objects C in Rd has the (l, k)-property if every subfamily C′ ⊆ C of cardinality at most l is k-piercable. Danzer and Gr¨unbaum posed the following fundamental question which can be considered as a generalised version of Helly’s theorem: For every positive integer k, does there exist a finite g(k, d) such that if any family of convex objects C in Rd has the (g(k, d), k)-property, then C is k-piercable? Very few results(mostly negative) are known. Inspired by the original question of Danzer and Gr¨unbaum we consider their question in an abstract set theoretic setting. Let U be a set (possibly infinite). Let C be a family of subsets of U with the property that if C1, . . . ,Cp+1 ∈ C are p + 1 distinct subsets, then |C1 ∩ · · · ∩Cp+1| ≤ l. In the second part of this thesis, we show in this setting, the first general positive results for the Danzer Grunbaum problem. As an extension, we show polynomial sized kernels for hitting set and covering problems in our setting. In the third part of this thesis, we broadly look at hitting and covering questions with respect to points and families of geometric objects in Rd. Let P be a subset of points(possibly infinite) in Rd and C be a collection of subsets of P induced by objects of a given family. For the system (P, C), let νh be the packing number and νc the dual packing number. We consider the problem of bounding the transversal number τ h and the dual transversal number τ c in terms of νh and νc respectively. These problems has been well studied in the case when P = R2. We systematically look at the case when P is finite, showing bounds for intervals, halfspaces, orthants, unit squares, skylines, rectangles, halfspaces in R3 and pseudo disks. We show bounds for rectangles when P = R2. Given a point set P ⊆ Rd, a family of objects C and a real number 0 < ǫ < 1, the strong epsilon net problem is to find a minimum sized subset Q ⊆ P such that any object C ∈ C with the property that |P ∩C| ≥ ǫn is hit by Q. It is customary to express the bound on the size of the set Q in terms of ǫ. Let G be a uniform √n × √n grid. It is an intriguing question as to whether we get significantly better bounds for ǫ-nets if we restrict the underlying point set to be the grid G. In the last part of this thesis we consider the strong epsilon net problem for families of geometric objects like lines and generalized parallelograms, when the underlying point set is the grid G. We also introduce the problem of finding ǫ-nets for arithmetic progressions and give some preliminary bounds.

Cette thèse est une contribution à un problème classique de la géométrie algorithmique et combinatoire : l’étude des k-sets d’un ensemble V de n points du plan. Nous introduisons d’abord la notion de chaîne d’inclusion de convexes qui est un ordonnancement des points de V tel qu’aucun point n’appartient à l’enveloppe convexe de ceux qui le précèdent. Tout k-set d’une sous-suite commençante de la chaîne est appelé un k-set de la chaîne. Nous montrons que le nombre de ces k-sets est un invariant de V et qu’il est égal au nombre de régions du diagramme de Voronoï d’ordre k de V. Nous en déduisons un algorithme en ligne pour construire les k-sets des sommets d’une ligne polygonale simple dont chaque sommet est à l’extérieur de l’enveloppe convexe des sommets qui le précèdent sur la ligne. Si c est le nombre total de k-sets construits, la complexité de notrealgorithme est en O(n log n+c log^2 k) et est équivalente, par k-set construit, à celle du meilleur algorithme connu. Nous montrons ensuite que la méthode algorithmique classique de division-fusion peut être adaptée à la construction des k-sets de V. L’algorithme qui en résulte a une complexité enO(n log n+c log^2 k log(n/k)), où c est le nombre maximum de k-sets d’un ensemble de n points.Nous prouvons enfin que les centres de gravité des k-sets d’une chaîne d’inclusion de convexes sont les sommets d’une triangulation qui appartient à la même famille de triangulations, dites centroïdes, que le dual du diagramme de Voronoï d’ordre k. Nous en d´déduisons un algorithme qui construit des triangulations centroïdes particulières en temps O(n log n+k(n-k) log^2 k), ce qui est plus efficace que les algorithmes connus jusque là. / This thesis is a contribution to a classical problem in computational and combinatorial geometry: the study of the k-sets of a set V of n points in the plane. First we introduce the notion of convex inclusion chain that is an ordering of the points of V such that no point is inside the convex hull of the points that precede it. Every k-set of an initial sub-sequence of the chain is called a k-set of the chain. We prove that the number of these k-sets is an invariant of V and is equal to the number of regions in the order-k Voronoi diagram of V. We then deduce an online algorithm for the construction of the k-sets of the vertices of a simple polygonal line such that every vertex of this line is outside the convex hull of all its preceding vertices on the line. If c is the total number of k-sets built with this algorithm, the complexity of our algorithm is in O(n log n + c log^2k) and is equal, per constructed k-set, to the complexity of the best algorithm known. Afterward, we prove that the classical divide and conquer algorithmic method can be adapted to the construction of the k-sets of V. The algorithm has a complexity of O(n log n + c log^2k log(n/k)), where c is the maximum number of k-sets of a set of n points. We finally prove that the centers of gravity of the k-sets of a convex inclusion chain are the vertices of a triangulation belonging to the family of so-called centroid triangulations. This family notably contains the dual of the order-k Voronoi diagram. We give an algorithm that builds particular centroid triangulations in O(n log n + k(n- k) log^2 k) time, which is more efficient than all the currently known algorithms.

This thesis mainly focuses on algorithmic and combinatorial questions related to some geometric problems on graphs. In the last part of this thesis, a graph coloring problem is also discussed. Boxicity and Cubicity: These are graph parameters dealing with geomet-ric representations of graphs in higher dimensions. Both these parameters are known to be NP-Hard to compute in general and are even hard to approximate within an O(n1− ) factor for any > 0, under standard complexity theoretic assumptions. We studied algorithmic questions for these problems, for certain graph classes, to yield efficient algorithms or approximations. Our results include a polynomial time constant factor approximation algorithm for computing the cubicity of trees and a polynomial time constant (≤ 2.5) factor approximation algorithm for computing the boxicity of circular arc graphs. As far as we know, there were no constant factor approximation algorithms known previously, for computing boxicity or cubicity of any well known graph class for which the respective parameter value is unbounded. We also obtained parameterized approximation algorithms for boxicity with various edit distance parameters. An o(n) factor approximation algorithm for computing the boxicity and cubicity of general graphs also evolved as an interesting corollary of one of these parameterized algorithms. This seems to be the first sub-linear factor approximation algorithm known for computing the boxicity and cubicity of general graphs. Planar grid-drawings of outerplanar graphs: A graph is outerplanar, if it has a planar embedding with all its vertices lying on the outer face. We give an efficient algorithm to 2-vertex-connect any connected outerplanar graph G by adding more edges to it, in order to obtain a supergraph of G such that the resultant graph is still outerplanar and its pathwidth is within a constant times the pathwidth of G. This algorithm leads to a constant factor approximation algorithm for computing minimum height planar straight line grid-drawings of outerplanar graphs, extending the existing algorithm known for 2-vertex connected outerplanar graphs. n−1 3 Maximum matchings in triangle distance Delaunay graphs: Delau-nay graphs of point sets are well studied in Computational Geometry. Instead of the Euclidean metric, if the Delaunay graph is defined with respect to the convex distance function defined by an equilateral triangle, it is called a Trian-gle Distance Delaunay graph. TD-Delaunay graphs are known to be equivalent to geometric spanners called half-Θ6 graphs. It is known that classical Delaunay graphs of point sets always contain a near perfect matching, for non-degenerate point sets. We show that Triangle Distance Delaunay graphs of a set of n points in general position will always l m contain a matching of size and this bound is tight. We also show that Θ6 graphs, a class of supergraphs of half-Θ6 graphs, can have at most 5n − 11 edges, for point sets in general position. Heterochromatic Paths in Edge Colored Graphs: Conditions on the coloring to guarantee the existence of long heterochromatic paths in edge col-ored graphs is a well explored problem in literature. The objective here is to obtain a good lower bound for λ(G) - the length of a maximum heterochro-matic path in an edge-colored graph G, in terms of ϑ(G) - the minimum color degree of G under the given coloring. There are graph families for which λ(G) = ϑ(G) − 1 under certain colorings, and it is conjectured that ϑ(G) − 1 is a tight lower bound for λ(G). We show that if G has girth is at least 4 log2(ϑ(G))+2, then λ(G) ≥ ϑ(G)− 2. It is also proved that a weaker requirement that G just does not contain four-cycles is enough to guarantee that λ(G) is at least ϑ(G) −o(ϑ(G)). Other special cases considered include lower bounds for λ(G) in edge colored bipartite graphs, triangle-free graphs and graphs without heterochromatic triangles.

In this thesis, we focus on the study of computational and combinatorial problems on various geometric proximity graphs. Delaunay and Gabriel graphs are widely studied geometric proximity structures. These graphs have been extensively studied for their applications in wireless networks. Motivated by the applications in localized wireless routing, relaxed versions of these graphs known as Locally Delaunay Graphs (LDGs) and Locally Gabriel Graphs(LGGs) were proposed. A geometric graph G=(V,E)is called a Locally Gabriel Graph if for every( u,v) ϵ E the disk with uv as diameter does not contain any neighbor of u or v in G. Thus, two edges (u, v) and(u, w)where u,v,w ϵ V conflict with each other if ∠uwv ≥ or ∠uvw≥π and cannot co-exist in an LGG. We propose another generalization of LGGs called Generalized locally Gabriel Graphs(GLGGs)in the context when certain edges are forbidden in the graph. For a given geometric graph G=(V,E), we define G′=(V,E′) as GLGG if G′is an LGG and E′⊆E. Unlike a Gabriel Graph ,there is no unique LGG or GLGG for a given point set because no edge is necessarily included or excluded. This property allows us to choose an LGG/GLGG that optimizes a parameter of interest in the graph. While Gabriel graphs are planar graphs, there exist LGGs with super linear number of edges. Also, there exist point sets where a Gabriel graph has dilation of Ω(√n)and there exist LGGs on the same point sets with dilation O(1). We study these graphs for various parameters like edge complexity(the maximum number of edges in these graphs),size of an independent set and dilation. We show that computing an edge maximum GLGG for a given problem instance is NP-hard and also APX-hard. We also show that computing an LGG on a given point set with minimum dilation is NP-hard. Then, we give an algorithm to verify whether a given geometric graph G=(V,E)is an LGG with running time O(ElogV+ V). We show that any LGG on n vertices has an independent set of size Ω(√nlogn). We show that there exists point sets with n points such that any LGG on it has dilation Ω(√n) that matches with the known upper bound. Then, we study some greedy heuristics to compute LGGs with experimental evaluation. Experimental evaluations for the points on a uniform grid and random point sets suggest that there exist LGGs with super-linear number of edges along with an independent set of near-linear size. Unit distance graphs(UDGs) are well studied geometric graphs. In this graph, an edge exists between two points if and only if the Euclidean distance between the points is unity. UDGs have been studied extensively for various properties most notably for their edge complexity and chromatic number. These graphs have also been studied for various special point sets most notably the case when the points are in convex position. Note that the UDGs form a sub class of the LGGs. UDGs/LGGs on convex point sets have O(nlogn) edges. The best known lower bound on the edge complexity of these graphs is 2n−7 when all the points are in convex position. A bipartite graph is called an ordered bipartite graph when the vertex set in each partition has a total order on its vertices. We introduce a family of ordered bipartite graphs with restrictions on some paths called path restricted ordered bi partite graphs (PRBGs)and show that their study is motivated by LGGs and UDGs on convex point sets. We show that a PRBG can be extracted from the UDGs/LGGs on convex point sets. First, we characterize a special kind of paths in PRBGs called forward paths, then we study some structural properties of these graphs. We show that a PRBG on n vertices has O(nlogn) edges and the bound is tight. It gives an alternate proof of O(nlogn)upper bound for the maximum number of edges in UDGs/LGGs on convex point sets. We study PRBGs with restrictions to the length of the forward paths and show an improved bound on the edge complexity when the length of the longest forward path is bounded. Then, we study the hierarchical structure amongst these graphs classes. Notably, we show that the class of UDGs on convex point sets is a strict sub class of LGGs on convex point sets.

On considère, en deux dimensions, une version euclidienne discrète de l’action d’Einstein-Hilbert, qui décrit la gravité en l’absence de matière. À l’intégration sur les géométries se substitue une sommation sur des surfaces triangulées aléatoires. Dans la limite physique de faible gravité, seules les triangulations planaires survivent. Leur limite en distribution, la carte brownienne, est une surface fractale continue dont l’importance dans le contexte de la gravité quantique en deux dimensions a été récemment précisée. Cet espace est interprété comme un espace-temps quantique, obtenu comme limite à grande échelle d’un ensemble statistique de surfaces discrètes aléatoires. En deux dimensions, on peut donc étudier les propriétés fractales de la gravité quantique via une approche discrète. Il est bien connu que les généralisations directes en dimensions supérieures échouent à produire des espace-temps quantiques aux propriétés adéquates : en dimension D>2, la limite en distribution des triangulations qui survivent dans la limite de faible gravité est l’arbre continu aléatoire, ou polymères branchés en physique. Si en deux dimensions on parvient aux mêmes conclusions en considérant non pas des triangulations, mais des surfaces discrètes aléatoires obtenues par recollements de 2p-gones, nous savons depuis peu que ce n’est pas toujours le cas en dimension D>2. L’apparition de nouvelles limites continues dans le cadre de théories de gravité impliquant des espaces discrets aléatoires reste une question ouverte. Nous étudions des espaces obtenus par recollements de blocs élémentaires, comme des polytopes à facettes triangulaires. Dans la limite de faible gravité, seuls les espaces qui maximisent la courbure moyenne survivent. Les identifier est cependant une tâche ardue dans le cas général, pour lequel les résultats sont obtenus numériquement. Afin d’obtenir des résultats analytiques, une coloration des (D-1)-cellules, les facettes, a été introduite. En toute dimension paire, on peut trouver des familles d’espaces discrets colorés de courbure moyenne maximale dans la classe d’universalité des arbres – convergeant vers l’arbre continu aléatoire, des cartes planaires – convergeant vers la carte brownienne, ou encore dans la classe de prolifération des bébé-univers. Cependant, ces résultats sont obtenus en raison de la simplicité de blocs élémentaires dont la structure uni ou bidimensionnelle ne rend pas compte de la riche diversité des blocs colorés en dimensions supérieures. Le premier objectif de cette thèse est donc d’établir des outils combinatoires qui permettraient une étude systématique des blocs élémentaires colorés et des espaces discrets qu’ils génèrent. Le principal résultat de ce travail est l’établissement d’une bijection entre ces espaces et des familles de cartes combinatoires, qui préserve l’information sur la courbure locale. Elle permet l’utilisation de résultats sur les surfaces discrètes et ouvre la voie à une étude systématique des espaces discrets en dimensions supérieures à deux. Cette bijection est appliquée à la caractérisation d’un certain nombre de blocs de petites tailles ainsi qu’à une nouvelle famille infinie. Le lien avec les modèles de tenseurs aléatoires est détaillé. Une attention particulière est donnée à la détermination du nombre maximal de (D-2)-cellules et de l’action appropriée du modèle de tenseurs correspondant. Nous montrons comment utiliser la bijection susmentionnée pour identifier les contributions à un tout ordre du développement en 1/N des fonctions à 2n points du modèle SYK coloré, et appliquons ceci à l’énumération des cartes unicellulaires généralisées – les espaces discrets obtenus par recollement d’un unique bloc élémentaire – selon leur courbure moyenne. Pour tout choix de blocs colorés, nous montrons comment réécrire la théorie d’Einstein-Hilbert discrète correspondante comme un modèle de matrices aléatoires avec traces partielles, dit représentation en champs intermédiaires. / In two dimensions, the Euclidean Einstein-Hilbert action, which describes gravity in the absence of matter, can be discretized over random triangulations. In the physical limit of small Newton's constant, only planar triangulations survive. The limit in distribution of planar triangulations - the Brownian map - is a continuum fractal space which importance in the context of two-dimensional quantum gravity has been made more precise over the last years. It is interpreted as a quantum continuum space-time, obtained in the thermodynamical limit from a statistical ensemble of random discrete surfaces. The fractal properties of two-dimensional quantum gravity can therefore be studied from a discrete approach. It is well known that direct higher dimensional generalizations fail to produce appropriate quantum space-times in the continuum limit: the limit in distribution of dimension D>2 triangulations which survive in the limit of small Newton's constant is the continuous random tree, also called branched polymers in physics. However, while in two dimensions, discretizing the Einstein-Hilbert action over random 2p-angulations - discrete surfaces obtained by gluing 2p-gons together - leads to the same conclusions as for triangulations, this is not always the case in higher dimensions, as was discovered recently. Whether new continuum limit arise by considering discrete Einstein-Hilbert theories of more general random discrete spaces in dimension D remains an open question.We study discrete spaces obtained by gluing together elementary building blocks, such as polytopes with triangular facets. Such spaces generalize 2p-angulations in higher dimensions. In the physical limit of small Newton's constant, only discrete spaces which maximize the mean curvature survive. However, identifying them is a task far too difficult in the general case, for which quantities are estimated throughout numerical computations. In order to obtain analytical results, a coloring of (D-1)-cells has been introduced. In any even dimension, we can find families of colored discrete spaces of maximal mean curvature in the universality classes of trees - converging towards the continuous random tree, of planar maps - converging towards the Brownian map, or of proliferating baby universes. However, it is the simple structure of the corresponding building blocks which makes it possible to obtain these results: it is similar to that of one or two dimensional objects and does not render the rich diversity of colored building blocks in dimensions three and higher.This work therefore aims at providing combinatorial tools which would enable a systematic study of the building blocks and of the colored discrete spaces they generate. The main result of this thesis is the derivation of a bijection between colored discrete spaces and colored combinatorial maps, which preserves the information on the local curvature. It makes it possible to use results from combinatorial maps and paves the way to a systematical study of higher dimensional colored discrete spaces. As an application, a number of blocks of small sizes are analyzed, as well as a new infinite family of building blocks. The relation to random tensor models is detailed. Emphasis is given to finding the lowest bound on the number of (D-2)-cells, which is equivalent to determining the correct scaling for the corresponding tensor model. We explain how the bijection can be used to identify the graphs contributing at any given order of the 1/N expansion of the 2n-point functions of the colored SYK model, and apply this to the enumeration of generalized unicellular maps - discrete spaces obtained from a single building block - according to their mean curvature. For any choice of colored building blocks, we show how to rewrite the corresponding discrete Einstein-Hilbert theory as a random matrix model with partial traces, the so-called intermediate field representation.