Extremal problems for ordered hypergraphs: small patterns and some enumeration

Extremal Problems (and a Bit of Enumeration) for Hypergraphs with Linearly Ordered Vertex Sets

On a Turán Problem in Weakly Quasirandom 3-uniform Hypergraphs

Extremal problems for 3-uniform hypergraphs are known to be very difficult and despite considerable effort the progress has been slow. We suggest a more systematic study of extremal problems in the context of quasirandom hypergraphs. We say that a 3-uniform hypergraph H “ pV,Eq is weakly pd, ηq-quasirandom if for any subset U Ď V the number of hyperedges of H contained in U is in the interval d...

Extremal problems for ordered (hyper)graphs: applications of Davenport-Schinzel sequences

We introduce a containment relation of hypergraphs which respects linear orderings of vertices and investigate associated extremal functions. We extend, by means of a more generally applicable theorem, the n log n upper bound on the ordered graph extremal function of F = ({1, 3}, {1, 5}, {2, 3}, {2, 4}) due to Füredi to the n(log n)2(log log n)3 upper bound in the hypergraph case. We use Davenp...

Analytic methods for uniform hypergraphs∗†

This paper presents some analytic methods for studying uniform hypergraphs. Its starting point is the spectral theory of 2-graphs, in particular, the largest and the smallest eigenvalues λ and λmin of 2-graphs. First, these two parameters are extended to weighted uniform hypergraphs; second, the eigenvalues-numbers λ and λmin are extended to eigenvalues-functions λ(p) and λ (p) min, which also ...

Structure and Enumeration Theorems for Hereditary Properties in Finite Relational Languages

Given a finite relational language L, a hereditary L-property is a class of finite Lstructures which is closed under isomorphism and model theoretic substructure. This notion encompasses many objects of study in extremal combinatorics, including (but not limited to) hereditary properties of graphs, hypergraphs, and oriented graphs. In this paper, we generalize certain definitions, tools, and re...