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Generalized Hyperfocused Arcs in P G ( 2 , p )
Author(s) -
Blokhuis Aart,
Marino Giuseppe,
Mazzocca Francesco
Publication year - 2014
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21371
Subject(s) - mathematics , arc (geometry) , conjecture , combinatorics , prime (order theory) , point (geometry) , section (typography) , property (philosophy) , cylinder , set (abstract data type) , geometry , philosophy , epistemology , computer science , programming language , advertising , business
A generalized hyperfocused arc H in P G ( 2 , q ) is an arc of size k with the property that the k ( k − 1 ) / 2 secants can be blocked by a set of k − 1 points not belonging to the arc. We show that if q is a prime and H is a generalized hyperfocused arc of size k , then k = 1 , 2 , or 4. Interestingly, this problem is also related to the (strong) cylinder conjecture ([2][S. Ball, 2012], [5][P. J. Cameron, 2008] Problem 919), as we point out in the last section.