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Leonardo N. Coregliano appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.
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Recently, a series of works have started studying variations of concepts from learning theory for product spaces, which can be collected under the name high-arity learning theory. In this work, we consider a high-arity variant of sample compression schemes and we prove that the existence of a high-arity sample compression scheme of non-trivial quality implies high-arity PAC learnability.
A Sidorenko bigraph is one whose density in a bigraphon $W$ is minimized precisely when $W$ is constant. Several techniques of the literature to prove the Sidorenko property consist of decomposing (typically in a tree decomposition) the bigraph into smaller building blocks with stronger properties. One prominent such technique is that of $N$-decompositions of Conlon--Lee, which uses weakly Hölder (or weakly norming) bigraphs as building blocks. In turn, to obtain weakly Hölder bigraphs, it is typical to use the chain of implications reflection bigraph $\implies$ cut-percolating bigraph $\implies$ weakly Hölder bigraph. In an earlier result by the author with Razborov, we provided a generalization of $N$-decompositions, called reflective tree decompositions, that uses much weaker building blocks, called induced-Sidorenko bigraphs, to also obtain Sidorenko bigraphs. In this paper, we show that "left-sided" versions of the concepts of reflection bigraph and cut-percolating bigraph yield a similar chain of implications: left-reflection bigraph $\implies$ left-cut-percolating bigraph $\implies$ induced-Sidorenko bigraph. We also show that under mild hypotheses, the "left-sided" analogue of the weakly Hölder property (which is also obtained via a similar chain of implications) can be used to improve bounds on another result of Conlon--Lee that roughly says that bigraphs with enough vertices on the right side of each realized degree have the Sidorenko property.
The celebrated Erdős--Stone--Simonovits theorem characterizes the asymptotic maximum edge density in $\mathcal{F}$-free graphs as $1 - 1/(χ(\mathcal{F})-1) + o(1)$, where $χ(\mathcal{F})$ is the minimum chromatic number of a graph in $\mathcal{F}$. In Examples 25 and 31 of [L. N. Coregliano and A. A. Razborov. Semantic limits of dense combinatorial objects. Uspekhi Mat. Nauk, 75(4(454)):45-152, 2020], it was shown that this result can be extended to the general setting of graphs with extra structure: the maximum asymptotic density of a graph with extra structure without some induced subgraphs is $1 - 1/(χ(I) - 1) + o(1)$ for an appropriately defined abstract chromatic number $χ(I)$. As the name suggests, the original formula for the abstract chromatic number is so abstract that its (algorithmic) computability was left open. In this paper, we both extend this result to characterize maximum asymptotic density of $t$-cliques in of graphs with extra structure without some induced subgraphs in terms of $χ(I)$ and we present a more concrete formula for $χ(I)$ that allows us to show its computability when both the extra structure and the forbidden subgraphs can be described by a finitely axiomatizable universal first-order theory. Our alternative formula for $χ(I)$ makes use of a partite version of Ramsey's Theorem for structures on first-order relational languages.
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.