Xiaofan Yuan

(Advisor: Prof. Xingxing Yu) will defend a doctoral thesis entitled,

Matching Problems in Hypergraphs

On

Thursday, June 9 at 10:00 a.m.

Skiles 006

https://gatech.zoom.us/j/91659544858?pwd=SWZtVG15dGFiWEFXSHR1U0JNbVVBZz09

Abstract

Ku¨hn, Osthus, and Treglown and, independently, Khan proved that if H is a 3-uniform hypergraph

with n vertices, where n ∈ 3Z and large, and δ₁(H) > ⁿ−¹ − 2n/3 , then H contains a perfect

We show that for n 3Z sufficiently large, if F , . . . , F are 3-uniform hypergraphs

with a common vertex set and δ₁(Fi) > (n−1)−(2n/3) for i ∈ [n/3], then {F₁, . . . , Fn/3} admits a

rainbow

matching, i.e., a matching consisting of one edge from each Fi. This is done by converting the

rainbow matching problem to a perfect matching problem in a special class of uniform hypergraphs.

We also prove that, for any integers k, l with k ≥ 3 and k/2 < l ≤ k − 1, there exists a positive

real µ such that, for all sufficiently large integers m, n satisfying

n

k

− µn ≤ m

≤

n

k

−

1

l

l

2l − k

,

if H is a k-uniform hypergraph on n vertices and δl(H) >

(n−

) −

) hen H has a

. This improves upon an earlier result of H`an, Person, and Schacht for the range k/2 < l ≤ k − 1.

In many cases, our result gives tight bound on δl(H) for near perfect matchings (e.g., when l ≥

2k/3, n ≡ r (mod k), 0 ≤ r < k, and r + l ≥ k, we can take m = n/k − 2).

Committee

• Prof. Anton Bernshteyn - School of Mathematics, Georgia Institute of Technology

• Prof. Hao Huang - Department of Mathematics, National University of Singapore (reader)

• Prof. Santosh Vempala - College of Computing, Georgia Institute of Technology

• Prof. Josephine Yu - School of Mathematics, Georgia Institute of Technology

• Prof. Xingxing Yu - School of Mathematics, Georgia Institute of Technology (advisor)