Ultrahomogeneous semilinear spaces (Q2766440)

Recall that a semilinear space \(S\) is a non-empty set of elements called points, provided with a collection of subsets called lines such that any pair of points is contained in at most one line and every line contains at least two points. NEWLINENEWLINENEWLINENote that semilinear spaces are a common generalization of graphs and of linear spaces. A semilinear space which is neither a graph nor a linear space is called proper. A semilinear space \(S\) is said to be \(d\)-homogeneous for some positive integer \(d\) if, whenever the semilinear structures induced on two subsets \(S_1\) and \(S_2\) of \(S\) of cardinality at most \(d\) are isomorphic, there is at least one automorphism of \(S\) mapping \(S_1\) onto \(S_2\); if every isomorphism from \(S_1\) to \(S_2\) extends to an automorphism of \(S\), then \(S\) is called \(d\)-ultrahomogeneous. A semilinear space \(S\) is called homogeneous if \(S\) is \(d\)-homogeneous for every positive integer \(d\), and it is ultrahomogeneous if \(S\) is \(d\)-ultrahomogeneous for every \(d\). The purpose of the paper is to prove the following result. NEWLINENEWLINENEWLINETheorem 1.0.1. If \(S\) is a \(6\)-ultrahomogeneous non-connected semilinear space, then \(S\) is ultrahomogeneous and the connected components of \(S\) are isomorphic ultrahomogeneous linear spaces. Any finite connected \(6\)-ultrahomogeneous semilinear space is ultrahomogeneous and is one of the following: (1) a graph \(C_5, L_{3,3}, K_n\) or \(K_{t;n}\) with \(t,n \geq 2\); (2) a single point or a single line; (3) one of the projective planes PG\((2,2)\) or PG\((2,3)\); (4) the \(3 \times 3\)-grid, i.e. the unique generalized quadrangle of order \((2,1)\); (5) one of the transversal designs TD\((3,2)\) or TD\((4,3)\); (6) \(U_{2,3}(n)\) for any integer \(n \geq 5\); (7) a triangular space \(T(n)\) for any integer \(n \geq 5\). NEWLINENEWLINENEWLINEActually, the author proves a slightly stronger result: any connected proper \(3\)-ultrahomogeneous semilinear space is a polar space, a copolar space or a partial geometry; any \(5\)-homogeneous proper polar space is ultrahomogeneous; any \(5\)-ultrahomogeneous proper connected copolar space is ultrahomogeneous; any \(4\)-ultrahomogeneous finite partial geometry (with parameters \(\alpha, \beta \geq 2\)) is ultrahomogeneous.