Line-closed Subsets of Steiner Triple Systems and Classical Linear Spaces 1
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چکیده
A proper non-empty subset C of the points of a linear space S = (P; L) is called line-closed if any two intersecting lines of S , each meeting C at least twice, have their intersection in C. We show that when every line has k points and every point lies on r lines the maximum size for such sets is r + k ? 2. In addition it is shown that this cannot happen for projective spaces PG(n; q) unless q = 2, nor can it be obtained for aane spaces AG(n; q) unless n = 2 and q = 3. However, for all odd values of r there exist Steiner triple systems having such maximum line-closed subsets.
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