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Maximal intersecting families and affine regular polygons in PG(2, q)

1989
*
Journal of combinatorial theory. Series A
*

A

doi:10.1016/0097-3165(89)90057-5
fatcat:b2dezeexhbggvchlpxj3cgtiwy
*family*of mutually*intersecting*k-sets is called a k-clique. A k-clique is*maximal*if it is not contained*in*any larger k-clique. ... Using a classification result of Wettl we give a new upper bound for m(k), the minimum number of members of a*maximal*k-clique, proving m(k)< k2/*2*+5k+o(k) whenever k-1 is a prime power. ... PRELIMINARIES ON*MAXIMAL*CLIQUES Let k be a positive integer. A k-clique (or*intersecting**family*of rank k) is a collection of pairwise nondisjoint k-sets. ...##
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Page 9109 of Mathematical Reviews Vol. , Issue 2002M
[page]

2002
*
Mathematical Reviews
*

These lines meet if

*and*only if the corresponding bishops attack. Using the above model, the authors construct more*families*of*maximal*partial spreads*in**PG*(3,*q*). ... The basic idea is to consider one line / of a*regular*spread*in**PG*(3,*q*)*and*a certain set of 5(*q*+1) mutu- ally disjoint reguli not containing /. ...##
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Contents

2003
*
European journal of combinatorics (Print)
*

.

doi:10.1016/s0195-6698(03)00149-5
fatcat:4nmkuxvgbzgr7nicrcrobdwvoa
*and*DOVER, J.M., Blocking semiovals*in**PG*(*2*, 7)*and*beyond . . . . . . . . . . . . . . . . . 183 BREŠAR, B.,*Intersection*graphs of*maximal*hypercubes . . . . . 195 DONATI, G.*and*DURANTE, N., ... small complete caps*in**PG*(n,*2*) . . . . . . . . . . . . . . . . 613 ZHANG, X. ...##
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Index

2003
*
European journal of combinatorics (Print)
*

tree-like equalities . . . . . . .
557
BROWN, M.R., Ovoids of

doi:10.1016/s0195-6698(03)00150-1
fatcat:q6zy3wos7vdkzfy55qdwge5nl4
*PG*(3,*q*) stabilized by a central collineation . 409 BROWN, M.R., DE BEULE, J.*and*STORME, L.,*Maximal*partial spreads of T*2*(O)*and*... J.M., Blocking semiovals*in**PG*(*2*, 7)*and*beyond . . . . . . . . . . . . . . . . 183 REIFEGERSTE, A., On the diagram of 132-avoiding permutations . . 759 SCHROEDER, W.C.*and*SUROWSKI, D.B.,*Regular*...##
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Page 3294 of Mathematical Reviews Vol. , Issue 89F
[page]

1989
*
Mathematical Reviews
*

The relationship with the

*intersecting*concept of “*affinely**regular**polygon*” [see Korchmaros, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. ... (*In*each case, the n-arc is contained*in*a conic*and*is such a*regular**polygon*if p? tn, where g =p", p prime.) ...##
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Generalized Quadrangles Of Order (s, s^2). IV. Translations, Moufang And Fong-Seitz
[article]

2022
*
arXiv
*
pre-print

*In*this Part IV we classify all finite translation generalized quadrangles of order (s, s^

*2*) having a kernel of size at least 3, containing a

*regular*line not incident with the translation point. ... There are several applications on generalized quadrangles of order (s, s^

*2*) having at least

*2*translation points, on Moufang quadrangles,

*and*concerning the theorem of Fong

*and*Seitz classifying all groups ... For n =

*2*the elements of the

*family*of

*maximal*(n − 1)-dimensional spaces of a S 1;n−1 ,

*and*for n =

*2*the elements of any of the two

*families*of lines of S 1;1 , form an (n − 1)-regulus of S 1;n−1 ; ...

##
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On generalizing generalized polygons

2009
*
Innovations in Incidence Geometry Algebraic Topological and Combinatorial
*

Indeed, the first two members of the series are exactly the

doi:10.2140/iig.2009.10.147
fatcat:gdlqvmctdrao7c22or3o4bukpq
*affine*parts of the generalized*polygons*of type A*2**and*B*2*. ... Construction of the graphs*in*question is motivated by the way*in*which*regular*generalized*polygons*may be embedded*in*their Lie algebras, so that point-line incidence corresponds to the vanishing Lie ... I am grateful to Jef Thas*and*Ernie Shult, each of whom expressed keen interest*in*the talk*and*encouraged that it be manifested*in*text. ...##
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Page 4578 of Mathematical Reviews Vol. , Issue 83k
[page]

1983
*
Mathematical Reviews
*

Ital. (3) 11 (1956), 248-252;
MR 18, 61] gave the following construction for a

*family*F of (*q*?+*q*+*2*)/*2*-caps of*PG*(3,*q*). Let*Q*be an elliptic quadric*and*P a point not on*Q*. ... Ferri, Osvaldo 83k:51015 On type ((*q*—3)/*2*,(*q*—1)/*2*,*q*— 1) k-sets*in*an*affine*plane Ay Combinatorial*and*geometric structures*and*their applications (Trento, 1980), pp. 211-218, North-Holland Math. ...##
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Finite geometries

2008
*
Designs, Codes and Cryptography
*

Sziklai, Small multiple blocking sets

doi:10.1007/s10623-007-9162-6
fatcat:yd5cqfmlrfa7zlzeifhhx7qfg4
*in**PG*(4,*q**2*) with respect to planes K. Thas, A Lenz-Barlotti classification for finite generalized quadrangles V.D. ... Lunardon, Spreads*in*H(*q*)*and*1-systems of*Q*(6,*q*) D. Luyckx, On 1-systems of*Q*(6,*q*),*q*even R. Meshulam, Group algebras*and*expanders K. Metsch, Large caps of the Klein quadric*2*A. ... Szőnyi) A t-fold blocking set B*in**PG*(*2*,*q*) is a set of points such that every line of*PG*(*2*,*q*)*intersects*B*in*at least t points. ...##
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Page 1714 of Mathematical Reviews Vol. , Issue 84e
[page]

1984
*
Mathematical Reviews
*

s4e:05001
Michele Crismale, (

*q*?+*q*+ 1)-sets of type (0, 1,2,*q*+ 1)*in*translation planes of order g* (pp. 225-228); F. De Clerck*and*J. A. Thas, The embedding of (0,a)-geometries*in**PG*(n,*q*). ...*Families*of finite sets*in*which no set is covered by the union of two others. J. Combin. Theory Ser. A 33 (1982), no.*2*, 158-166. ...##
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Page 288 of Mathematical Reviews Vol. , Issue 89A
[page]

1989
*
Mathematical Reviews
*

Another con- struction uses

*families*of conics*and*leads to n <*2*(*q*—1)/s+1 for $|\(*q*— 1)*in**PG*(*2*, g). An analogous construction is given*in**PG*(3,*q*) via elliptic quadrics. ... Wagner*and*the reviewer has shown that*regular*sets*in**PG*(d,*q*) or AG(d,*q*) always exist unless both d*and**q*are very small. ...##
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Class fields of abelian extensions of Q

1984
*
Inventiones Mathematicae
*

Let R be a nontrivial local Z p-algebra of finite cardinality with

doi:10.1007/bf01388599
fatcat:7tu25wekq5atnaa7jfjv2qpopm
*maximal*ideal denoted m R . Then the following conditions are equivalent. ... But the integral closure alluded to is just the ring of*regular*functions on the*affine*scheme Y~ (N)/zKN][1/N ]*and*therefore we may view the Siegel functions as*regular*functions on Y1 (N)/z tCul [1/ ... Let E* be a generalized elliptic curve over S*in*the sense developed*in*([15,I1); we require that E* be a proper flat S-scheme whose geometric fibers are either elliptic curves or "*polygons**2*. ...##
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Page 4375 of Mathematical Reviews Vol. , Issue 2001G
[page]

2001
*
Mathematical Reviews
*

Taylor, Constructions for octonion

*and*exceptional Jordan al- gebres (191-203); Akos Seress, Large*families*of cospectral graphs (205-208); Ron Shaw, Subsets of*PG*(n,*2*)*and**maximal*partial spreads*in**PG*... Ray- Chaudhuri, New linear codes over F;*and*Fs*and*improvements on bounds (223-233); L. Storme*and*Zs. Weiner, On 1-blocking sets*in**PG*(n,*q*), n > 3 (235-251). ...##
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Near polygons and Fischer spaces

1994
*
Geometriae Dedicata
*

*In*this paper we construct some infinite

*families*of near

*polygons*,

*and*classify near hexagons with lines of length 3

*and*with quads. ... We also construct some infinite

*families*of near

*polygons*. ... Each quad from one

*family*meets each quad from the other

*family*

*in*a line. (The corresponding Fischer space is the disjoint union of two

*affine*planes of order 3.) ...

##
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Tight sets and m-ovoids of generalised quadrangles

2009
*
Combinatorica
*

a nondegenerate conic

doi:10.1007/s00493-009-2179-x
fatcat:3lv4kl6qafbfdemivsj73n6miq
*in*the plane*PG*(*2*,*q*),*q*odd, by means of their*intersection*pattern with lines. ... On*regular*hyperbolic fibrations Deirdre Luyckx A hyperbolic fibration of*PG*(3,*q*) is a collection of*q*− 1 hyperbolic quadrics*Q*+ i (3,*q*), i = 1,*2*, . . . ,*q*− 1,*and*two lines L 0 , L ∞*in**PG*(3,*q*...
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