Linearity of Saturation for Berge Hypergraphs

For a graph F, we say a hypergraph H is Berge-F if it can be obtained from F be replacing each edge of F with a hyperedge containing it. We say a hypergraph is Berge-F-saturated if it does not contain a Berge-F, but adding any hyperedge creates a copy of Berge-F. The k-uniform saturation number of Berge-F, satk(n, Berge-F) is the fewest number of edges in a Berge-F-saturated k-uniform hypergraph on n vertices. We show that satk(n, Berge-F) = O(n) for all graphs F and uniformities 3 ≤ k ≤ 5, partially answering a conjecture of English, Gordon, Graber, Methuku, and Sullivan. We also extend this conjecture to Berge copies of hypergraph

Saturation Numbers for Berge Cliques

Let $F$ be a graph and $\mathcal{H}$ be a hypergraph, both embedded on the same vertex set. We say $\mathcal{H}$ is a Berge-$F$ if there exists a bijection $\phi:E(F)\to E(\mathcal{H})$ such that $e\subseteq \phi(e)$ for all $e\in E(F)$. We say $\mathcal{H}$ is Berge-$F$-saturated if $\mathcal{H}$ does not contain any Berge-$F$, but adding any missing edge to $\mathcal{H}$ creates a copy of a Berge-$F$. The saturation number $\mathrm{sat}_k(n,\text{Berge-}F)$ is the least number of edges in a Berge-$F$-saturated $k$-uniform hypergraph on $n$ vertices. We show $$\mathrm{sat}_k(n,\text{Berge-}K_\ell)\sim \frac{\ell-2}{k-1}n,$$ for all $k,\ell\geq 3$. Furthermore, we provide some sufficient conditions to imply that $\mathrm{sat}_k(n,\text{Berge-}F)=O(n)$ for general graphs $F$.Comment: 16 pages, 1 figur

On the complexity of enumerating pseudo-intents

AbstractWe investigate whether the pseudo-intents of a given formal context can efficiently be enumerated. We show that they cannot be enumerated in a specified lexicographic order with polynomial delay unless P=NP. Furthermore we show that if the restriction on the order of enumeration is removed, then the problem becomes at least as hard as enumerating minimal transversals of a given hypergraph. We introduce the notion of minimal pseudo-intents and show that recognizing minimal pseudo-intents is polynomial. Despite their less complicated nature, surprisingly it turns out that minimal pseudo-intents cannot be enumerated in output-polynomial time unless P=NP

Combinatorial methods in differential algebra

This thesis studies various aspects of differential algebra, from fundamental concepts to practical computations. A characteristic feature of this work is the use of combinatorial techniques, which offer a unique and original perspective on the subject matter. First, we establish the connection between the n-jet space of the fat point defined by xm and the stable set polytope of a perfect graph. We prove that the dimension of the coordinate ring of the scheme defined by polynomial arcs of degree less than or equal to n is a polynomial in m of degree n + 1. This is based on Zobnin’s result which states that the set {x^m} is a differential Gr ̈obner basis for its differential ideal. We generalize this statement to the case of two independent variables and link the dimensions in this case to some triangulations of the p × q rectangle, where the pair (p, q) now plays the role of n. Second, we study the arc space of the fat point x^m on a line from the point of view of filtration by finite-dimensional differential algebras. We prove that the generating series of the dimensions of these differential algebras is m/(1 -mt) . Based on this we propose a definition of the multiplicity of a solution of an algebraic differential equation as the growth of the dimensions of these differential algebras. This generalizes the concept of the multiplicity of an ideal in a polynomial ring. Furthermore, we determine a full description of the set of standard monomials of the differential ideal generated by x^m. This description proves a conjecture by Afsharijoo concerning a new version of the Roger-Ramanujan identities. Every homogeneous linear system of partial differential equations with constant coef- ficients can be encoded by a submodule of the ring of polynomials. We develop practical methods for computing the space of solutions to these PDEs. These spaces are typically infinite dimensional, and we use the Ehrenpreis–Palamodov Theorem for finite encoding. We apply this finite encoding to the solutions of the PDEs associated with the arc spaces of a double point. We prove that these vector spaces are spanned by determinants of some special Wronskians, and we relate them to differentially homogeneous polynomials. Finally, we introduce D-algebraic functions: they are solutions to algebraic differential equations. We study closure properties of these functions. We present practical algorithms and their implementations for carrying out arithmetic operations on D-algebraic functions. This amounts to solving elimination problems for differential ideals