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Title: Linear Hypothesis Testing in Linear Models With High-Dimensional Responses
Award ID(s):
1820702 1953196 2015539
PAR ID:
10436268
Author(s) / Creator(s):
Date Published:
Journal Name:
Journal of the American Statistical Association
Volume:
117
Issue:
540
ISSN:
0162-1459
Page Range / eLocation ID:
1738 to 1750
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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  2. Abstract A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in ann-vertex graph not containingC2k, the cycle of length 2k, isO(n1+1/k). Simonovits established a corresponding supersaturation result forC2k’s, showing that there exist positive constantsC,cdepending only onksuch that everyn-vertex graphGwithe(G)⩾Cn1+1/kcontains at leastc(e(G)/v(G))2kcopies ofC2k, this number of copies tightly achieved by the random graph (up to a multiplicative constant). In this paper we extend Simonovits' result to a supersaturation result ofr-uniform linear cycles of even length inr-uniform linear hypergraphs. Our proof is self-contained and includes ther= 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest. 
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