Nika Salia (Alfréd Rényi Institute of Mathematics, Central European University): The Last and Maybe The Most Interesting Case of Erdős and Gallai Theorem for Berge Hypergraphs

A well-known theorem of Erdős and Gallai from 1959, asserts that a graph

with no path of length k contains at most $\frac{(k−1)n}{2}$ edges and a graph with no

cycle of length at least k contains at most $\frac{(k-1)(n-1)}{2}$ edges. Extensions of this

results for hypergraphs recently got very well studied. We are investigating one special

case when the forbidden path length is same as uniformity. This is the only missing case

for Berge Hypergraphs and in some sense the most interesting, since the extremal

hypergraph contractions are different from already existing ones. Proof is short and

simple and even more the same proof works for all path lengths less than uniformity.