By Michael Huber

On account of the category of the finite basic teams, it hasbeen attainable in recent times to signify Steiner t-designs, that's t -(v, okay, 1) designs,mainly for t = 2, admitting teams of automorphisms with sufficiently strongsymmetry homes. even if, regardless of the finite basic workforce class, forSteiner t-designs with t > 2 each one of these characterizations have remained longstandingchallenging difficulties. in particular, the decision of all flag-transitiveSteiner t-designs with three ≤ t ≤ 6 is of specific curiosity and has been open for about40 years (cf. Delandtsheer (Geom. Dedicata forty-one, p. 147, 1992 and instruction manual of IncidenceGeometry, Elsevier technology, Amsterdam, 1995, p. 273), yet most likely datingback to 1965).The current paper keeps the author's paintings (see Huber (J. Comb. thought Ser.A ninety four, 180-190, 2001; Adv. Geom. five, 195-221, 2005; J. Algebr. Comb., 2007, toappear)) of classifying all flag-transitive Steiner 3-designs and 4-designs. We provide acomplete class of all flag-transitive Steiner 5-designs and turn out furthermorethat there aren't any non-trivial flag-transitive Steiner 6-designs. either effects depend on theclassification of the finite 3-homogeneous permutation teams. in addition, we surveysome of the main common effects on hugely symmetric Steiner t-designs.

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**Extra resources for A census of highly symmetric combinatorial designs**

**Sample text**

Given n E N, what is the most likely number of summands in a random partition of n? Writing ko(n) for this number, it is not clear that ko(n) is welldefined although, as was shown later by Szekeres, this is the case. However, what does seem to be clear is that ko(n) is about A(n). Erdos proved that, in fact, , = A(n) + --;v'6 (log --;v'6). vn + o( vn). ko(n) Another circle of problems that has occupied Erdos for over sixty years originated with a question raised by Sidon when Erdos and Turan went to see him.

18 Bela Bollobas In 1953 Erdos returned to this theme. In a joint paper with Hunt he proved that if X l ,X2 , ... are independent zero-mean random variables with the same continuous distribution which is symmetric about 0 then, almost surely, Il· m -1 n-oc lorrn b 2: signSk n k=l 0 ----. k In his joint papers with Dvoretzky, Kac and Kakutani, Erdos contributed much to the theory of random walks and Brownian motion. For example, in 1940, Paul Levy proved that almost all paths of a Brownian motion in the plane have double points.

Graphs with slightly more edges than the extremal graph. An unpublished result of Rademacher from 1941 claims that a graph of order n with more than ln z /4J = t2(n) edges contains not only one triangle but at least In/2J triangles. In 1962 Erdos extended this result considerably; he showed that for some constant c> 0 every graph with n vertices and Ln 2 /4J + k edges has at least kln/2J triangles, provided 0 ::; k ::; en. Later, this led to a spate of related results by Erdos himself, Moon and Moser, Lovasz and Simonovits, Bollobas and others.

### A census of highly symmetric combinatorial designs by Michael Huber

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