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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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On the order of the classical Erdős–Rogers functions
Abstract For an integer , the Erdős–Rogers function is the maximum integer such that every ‐vertex ‐free graph has a ‐free induced subgraph with vertices. It is known that for all , as . In this paper, we show that for all , there exists a constant such thatThis improves previous bounds of order by Dudek, Retter and Rödl and answers a question of Warnke.
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- Award ID(s):
- 2347832
- PAR ID:
- 10644735
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- Volume:
- 57
- Issue:
- 2
- ISSN:
- 0024-6093
- Format(s):
- Medium: X Size: p. 582-598
- Size(s):
- p. 582-598
- Sponsoring Org:
- National Science Foundation
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