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=SWZtVG15dGFiWEFXSHR1U0JNbVVBZz…

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)