Semantic limits of dense combinatorial objects

Abstract

The theory of limits of discrete combinatorial objects has been thriving for the last decade or so. The syntactic, algebraic approach to the subject is popularly known as ‘flag algebras’, while the semantic, geometric approach is often associated with the name ‘graph limits’. The language of graph limits is generally more intuitive and expressible, but a price that one has to pay for it is that it is better suited for the case of ordinary graphs than for more general combinatorial objects. Accordingly, there have been several attempts in the literature, of varying degree of generality, to define limit objects for more complicated combinatorial structures. This paper is another attempt at a workable general theory of dense limit objects. Unlike previous efforts in this direction (with the notable exception of [5] by Aroskar and Cummings), our account is based on the same concepts from first-order logic and model theory as in the theory of flag algebras. It is shown how our definitions naturally encompass a host of previously considered cases (graphons, hypergraphons, digraphons, permutons, posetons, coloured graphs, and so on), and the fundamental properties of existence and uniqueness are extended to this more general case. Also given is an intuitive general proof of the continuous version of the Induced Removal Lemma based on the compactness theorem for propositional calculus. Use is made of the notion of open interpretation that often allows one to transfer methods and results from one situation to another. Again, it is shown that some previous arguments can be quite naturally framed using this language. Bibliography: 68 titles.

Polynomial to exponential transition in Ramsey theory

“…The edge coloring and the orientation between are read from the first coordinate in which . Further, is equipped with the measure that is the product of uniform measures on and all this structure turns into a ‐on ([8, Definition 3.2]). Hence we also have ([8, Theorem 6.3]) the corresponding algebra homomorphism ; its values are computed as obvious integrals over .…”

“…Further, is equipped with the measure that is the product of uniform measures on and all this structure turns into a ‐on ([8, Definition 3.2]). Hence we also have ([8, Theorem 6.3]) the corresponding algebra homomorphism ; its values are computed as obvious integrals over . In particular, is given by the ‘expected’ formula Along with Theorem 3.1, this leads us, after a bit of manipulations, to the bound When is a power of , the right‐hand side here is exactly (by an obvious induction).…”

Natural quasirandomness properties

“…We will be working in the framework of [CR20], in which combinatorial objects are encoded as models of a canonical theory. We will also be using the same notation as in [CR20] with some small additions.…”

“…First and foremost, we view this paper as a continuation of [Raz07,CR20], which in particular implies that we require qualifying properties to be formulated in an uniform way for arbitrary universal theories in a finite relational language. For examples of what can be expressed in that language see [CR20, Sct.…”

“…We have deliberately decided against attempting to state our properties in the language of finite combinatorial objects and their asymptotic behavior -it is probably possible but the result might be rather ugly and disappointing. Instead, we use the language of graphons [LS06], hypergraphons [ES12] and theons [CR20] for the geometric view of our objects and that of flag algebras [Raz07] for a concise algebraic description. We remark that we are not the first authors to make this election, and the advantages of using the continuous setting are illustrated by the fact that such proofs are often more elegant and less technical than their finite world counterparts [Jan11, KP13,Tow17].…”