# Combinatorics: Lior Gishboliner (TAU) "A Generalized Turan Problem and Its Applications"

Date:
Mon, 04/06/201811:00-12:30
Location:
IIAS, room 130, Feldman Building, Givat Ram
Speaker:   Lior Gishboliner, Tel Aviv University
Title:     A Generalized Turan Problem and Its Applications
Abstract:
The investigation of conditions guaranteeing the appearance of cycles
of certain lengths is one of the most well-studied topics in graph
theory. In this paper we consider a problem of this type which asks,
for fixed integers ℓ and k, how many copies of the k-cycle guarantee
the appearance of an ℓ-cycle? Extending previous results of
Bollobas-Gyori-Li and Alon-Shikhelman, we fully resolve this problem
by giving tight (or nearly tight) bounds for all values of ℓ and k.

We also present a somewhat surprising application of the above mentioned
estimates to the study of the graph removal lemma. Prior to this work,
all bounds for removal lemmas were either polynomial or there was a
tower-type gap between the best known upper and lower bounds. We fill
this gap by showing that for every super-polynomial function f(ε), there
is a family of graphs F, such that the bounds for the F removal lemma
are precisely given by f(ε). We thus obtain the first examples of
removal lemmas with tight super-polynomial bounds. A special case of
this result resolves a problem of Alon-Shapira, while another special
case partially resolves a problem of Goldreich.

Joint work with Asaf Shapira