Combinatorics and Discrete Mathematics Seminar: Survey of Recent Generalisations of Erd\H{o}s--Gallai Theorems for Berge Hypergraphs

Speaker: Nika Salia (Renyi Institute, Budapest)

Title: Survey of Recent Generalisations of Erd\H{o}s--Gallai Theorems for Berge Hypergraphs

Abstract:

A problem, first considered by Erd\H{o}s and Gallai in 1959, was to determine Tur\'an number of paths and families of long cycles. In particular Erd\H{o}s and Gallai determined the maximum number of edges in a graph without a path (not necessarily induced) of length $k$, $ex(n,P_k)\leq \frac{(k - 1)n}{2}$ for every  $n \geq k \geq 1$ and the maximum number of edges in a graph without a cycle of length at least $k$, $ex(n,C_{\geq k}) \leq \frac{(k - 1)(n - 1)}{2}$ for any $n \geq k \geq 3$.

Recently numerous mathematicians started investigating similar problems for $r$-uniform hypergraphs. They determined the maximum number of hyperedges in an $r-$uniform hypergraphs without  Berge paths/cycles. A Berge-path of length $k$ in a hypergraph is a sequence

$v_1,e_1,v_2,e_2,\dots,v_{k},e_k,v_{k+1}$ of distinct vertices and hyperedges with $v_{i+1}\in e_i,e_{i+1}$ for all $i\in[k]$.

In this talk we will try to survey those results as well as give you some ideas for possible further research.