Geodesic
Theorems of Helly--Gallai's type
Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 4, pp. 1209-1226.

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In this paper different generalizations of Helly's theorem for families of sets defined by systems of equations are presented. We investigate the existence problem of a $k$-element set, which has non-empty intersection with any member of such family. Applications to combinatorics and combinatorial geometry are given. 
@article{FPM_1999_5_4_a15,
     author = {V. L. Dol'nikov and S. A. Igonin},
     title = {Theorems of {Helly--Gallai's} type},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {1209--1226},
     publisher = {mathdoc},
     volume = {5},
     number = {4},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_1999_5_4_a15/}
}
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V. L. Dol'nikov; S. A. Igonin. Theorems of Helly--Gallai's type. Fundamentalʹnaâ i prikladnaâ matematika, Tome 5 (1999) no. 4, pp. 1209-1226. http://geodesic.mathdoc.fr/item/FPM_1999_5_4_a15/