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.

Abstract:We prove that for every fixed k, the number of occurrences of the transitive tournament Tr k of order k in a tournament T n on n vertices is asymptotically minimized when T n is random. In the opposite direction, we show that any sequence of tournaments {T n } achieving this minimum for any fixed k 4 is necessarily quasirandom. We present several other characterizations of quasirandom tournaments nicely complementing previously known results and relatively easily following from our proof techniques. C 2016 Wiley Periodicals, Inc. J. Graph Theory 00: 1-10, 2016

A tournament is called locally transitive if the outneighborhood and the inneighborhood of every vertex are transitive. Equivalently, a tournament is locally transitive if it avoids the tournaments W4 and L4, which are the only tournaments up to isomorphism on four vertices containing a unique 3‐cycle. On the other hand, a sequence of tournaments false(Tnfalse)n∈double-struckN with Vfalse(Tnfalse)=n is called almost balanced if all but o(n) vertices of Tn have outdegree (1/2+o(1))n. In the same spirit of quasi‐random properties, we present several characterizations of tournament sequences that are both almost balanced and asymptotically locally transitive in the sense that the density of W4 and L4 in Tn goes to zero as n goes to infinity.

We study the density of fixed strongly connected subtournaments on 5 vertices in large tournaments. We determine the maximum density asymptotically for five tournaments as well as unique extremal sequences for each tournament. As a byproduct we also characterize tournaments that are recursive blow-ups of a 3-cycle as tournaments that avoid three specific tournaments of size 5.

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