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

Endre Boros, Zoltán Füredi, Jeff Kahn
1989 Journal of combinatorial theory. Series A  
A 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.  ... 
doi:10.1016/0097-3165(89)90057-5 fatcat:b2dezeexhbggvchlpxj3cgtiwy

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 /.  ... 


2003 European journal of combinatorics (Print)  
. 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.  ... 
doi:10.1016/s0195-6698(03)00149-5 fatcat:4nmkuxvgbzgr7nicrcrobdwvoa


2003 European journal of combinatorics (Print)  
tree-like equalities . . . . . . . 557 BROWN, M.R., Ovoids of 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  ... 
doi:10.1016/s0195-6698(03)00150-1 fatcat:q6zy3wos7vdkzfy55qdwge5nl4

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.)  ... 

Generalized Quadrangles Of Order (s, s^2). IV. Translations, Moufang And Fong-Seitz [article]

Joseph A. Thas
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 ;  ... 
arXiv:2205.15260v1 fatcat:jgt6lcphmbgkbniblgrbz6gqey

On generalizing generalized polygons

Andrew Woldar
2009 Innovations in Incidence Geometry Algebraic Topological and Combinatorial  
Indeed, the first two members of the series are exactly the 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.  ... 
doi:10.2140/iig.2009.10.147 fatcat:gdlqvmctdrao7c22or3o4bukpq

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.  ... 

Finite geometries

A. Blokhuis, J. W. P. Hirschfeld, D. Jungnickel, J. A. Thas
2008 Designs, Codes and Cryptography  
Sziklai, Small multiple blocking sets 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.  ... 
doi:10.1007/s10623-007-9162-6 fatcat:yd5cqfmlrfa7zlzeifhhx7qfg4

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.  ... 

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.  ... 

Class fields of abelian extensions of Q

B. Mazur, A. Wiles
1984 Inventiones Mathematicae  
Let R be a nontrivial local Z p-algebra of finite cardinality with 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.  ... 
doi:10.1007/bf01388599 fatcat:7tu25wekq5atnaa7jfjv2qpopm

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).  ... 

Near polygons and Fischer spaces

A. E. Brouwer, A. M. Cohen, H. A. Wilbrink, J. J. Hall
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.)  ... 
doi:10.1007/bf01264034 fatcat:r2xtudd2g5hqdae6z56ags5mry

Tight sets and m-ovoids of generalised quadrangles

John Bamberg, Maska Law, Tim Penttila
2009 Combinatorica  
a nondegenerate conic 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  ... 
doi:10.1007/s00493-009-2179-x fatcat:3lv4kl6qafbfdemivsj73n6miq
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