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ABSTRACT For ak‐uniform hypergraph and a positive integer , the Ramsey number denotes the minimum such that every ‐vertex ‐free ‐uniform hypergraph contains an independent set of vertices. A hypergraph isslowly growingif there is an ordering of its edges such that for each . We prove that if is fixed and is any non‐k‐partite slowly growing ‐uniform hypergraph, then for ,In particular, we deduce that the off‐diagonal Ramsey number is of order , where is the triple system . This is the only 3‐uniform Berge triangle for which the polynomial power of its off‐diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs and hypergraph containers.
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On the fixed part of pluricanonical systems for surfaces
Abstract We show that defines a birational map and has no fixed part for some bounded positive integermfor any ‐lc surfaceXsuch that is big and nef. For every positive integer , we construct a sequence of projective surfaces , such that is ample, for everyi, , and for any positive integerm, there existsisuch that has nonzero fixed part. These results answer the surface case of a question of Xu.
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- PAR ID:
- 10531077
- Publisher / Repository:
- Wiley-VCH
- Date Published:
- Journal Name:
- Mathematische Nachrichten
- Volume:
- 296
- Issue:
- 5
- ISSN:
- 0025-584X
- Page Range / eLocation ID:
- 2046 to 2069
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1002/mana.202200088
