An easier proof of the maximal arcs conjecture

Ball, Simeon; Blokhuis, Aart

doi:10.1090/S0002-9939-98-04653-X

An easier proof of the maximal arcs conjecture
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by Simeon Ball and Aart Blokhuis

Proc. Amer. Math. Soc. 126 (1998), 3377-3380

DOI: https://doi.org/10.1090/S0002-9939-98-04653-X

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Abstract:

It was a long-standing conjecture in finite geometry that a Desarguesian plane of odd order contains no maximal arcs. A rather inaccessible and long proof was given recently by the authors in collaboration with Mazzocca. In this paper a new observation leads to a greatly simplified proof of the conjecture.

References

S. Ball, A. Blokhuis and F. Mazzocca, Maximal arcs in Desarguesian planes of odd order do not exist, Combinatorica 17, (1997), 31–41.
Aldo Cossu, Su alcune proprietà dei $\{k,\,n\}$-archi di un piano proiettivo sopra un corpo finito, Rend. Mat. e Appl. (5) 20 (1961), 271–277 (Italian). MR 139995
R. H. F. Denniston, Some maximal arcs in finite projective planes, J. Combinatorial Theory 6 (1969), 317–319. MR 239991, DOI 10.1016/S0021-9800(69)80095-5
László Rédei, Lückenhafte Polynome über endlichen Körpern, Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 42, Birkhäuser Verlag, Basel-Stuttgart, 1970 (German). MR 0294297, DOI 10.1007/978-3-0348-4006-4
J. A. Thas, Construction of maximal arcs and partial geometries, Geometriae Dedicata 3 (1974), 61–64. MR 349437, DOI 10.1007/BF00181361
J. A. Thas, Some results concerning $\{(q+1)(n-1);\,n\}$-arcs and $\{(q+1)$ $(n-1)+1;\,n\}$-arcs in finite projective planes of order $q$, J. Combinatorial Theory Ser. A 19 (1975), no. 2, 228–232. MR 377682, DOI 10.1016/s0097-3165(75)80012-4
J. A. Thas, Construction of maximal arcs and dual ovals in translation planes, European J. Combin. 1 (1980), no. 2, 189–192. MR 587531, DOI 10.1016/S0195-6698(80)80052-7