# Seminars & Colloquia Calendar

Discrete Math

## "Extremal problems for paths in graphs and hypergraphs"

#### Jacques Verstraete, UC San Diego

Location:  HILL 705
Date & time: Monday, 24 April 2017 at 2:00PM - 3:00PM

 Time: 2:00 PM Location: Hill 705 Abstract: The ErdH{o}s--Gallai Theorem states that any $$n$$-vertex graph without a $$k$$-edge path has at most $$frac{1}{2}(k-1)n$$ edges. In this talk, various generalizations of this theorem for graphs and hypergraphs will be discussed, using a number of novel combinatorial, geometric and probabilistic methods. A representative of our results is the following generalization of the ErdH{o}s--Gallai Theorem. A {em tight $$k$$-path} is the hypergraph comprising edges $${i,i + 1,dots,i + r - 1}$$, where $$1 leq i leq k$$. We give a short proof that if $$H$$ is an $$r$$-uniform $$n$$-vertex hypergraph not containing a tight $$k$$-path, then [ |H| leq frac{k-1}{r}{n choose r - 1}.] As noted by Kalai, equality holds whenever certain Steiner systems exist. This result proves a conjecture of Kalai for tight paths. We conclude with a number of open problems. One particular open problem, posed by V. S'{o}s and the speaker at AIM, is whether the maximum number of edges in an $$r$$-uniform $$n$$-vertex hypergraph containing no {em tight cycle} is at most $${n - 1 choose r - 1}$$. Joint work with Z. F"{u}redi, T. Jiang, A. Kostochka and D. Mubayi.

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