Zur Krümmungsverwandtschaft Von Zwangläufen in Affinen CK — Ebenen I

We study a map of osculating elements of an affine Cayley- Klein (CK-) plane into the Lie algebraA ₄(²) of the aequiform transformationsA ₄(²) of the given plane. If we use the real projective spaceP ₃() overA ₄(²) each osculating element defines a straight line inP ₃(). In the first part of this paper this map is studied in detail. In the second part we study second order properties of one- parameter motions and their corresponding properties in the Lie algebraA ₄(²). This is done by considering the analogen to the formula of EULERSAVARY in the image spaceP ³() overA ₄(²).     

Ramsey problem on multiplicities of complete subgraphs in nearly quasirandom graphs

Letk _t(G) be the number of cliques of ordert in the graphG. For a graphG withn vertices let \(c_t (G) = \frac{{k_t (G) + k_t (\bar G)}}{{\left( {\begin{array}{*{20}c} n \\ t \\ \end{array} } \right)}}\) . Letc _t(n)=Min{c _t(G)∶?G?=n} and let \(c_t = \mathop {\lim }\limits_{n \to \infty } c_t (n)\) . An old conjecture of Erdös [2], related to Ramsey's theorem states thatc _t=2^1-(t/2). Recently it was shown to be false by A. Thomason [12]. It is known thatc _t(G)≈2^1-(t/2) wheneverG is a pseudorandom graph. Pseudorandom graphs — the graphs “which behave like random graphs” — were inroduced and studied in [1] and [13]. The aim of this paper is to show that fort=4,c _t(G)≥2^1-(t/2) ifG is a graph arising from pseudorandom by a small perturbation.     

The strong fixed point property for lexicographic sums

The following theorem is proved: If Q=L{P _t tT} is a finite lexicographic sum of posets such that T and all P _t have the strong fixed point property then Q has the strong fixed point property. Moreover we show the strong fixed point property for two more classes of posets.