329 research outputs found
Short proofs of some extremal results
We prove several results from different areas of extremal combinatorics,
giving complete or partial solutions to a number of open problems. These
results, coming from areas such as extremal graph theory, Ramsey theory and
additive combinatorics, have been collected together because in each case the
relevant proofs are quite short.Comment: 19 page
On the seven non-isomorphic solutions of the fifteen schoolgirl problem
In this paper we give a simple and effective tool to analyze a given Kirkman triple system of order 15 and determine which of the seven well-known non-isomorphic KTS(15)s it is isomorphic to. Our technique refines and improves the lacing of distinct parallel classes introduced by F. N. Cole, by means of the notion of residual triple defined by G. Falcone and the present author in a previous paper. Unlike Cole's original lacing scheme, our algorithm allows one to distinguish two KTS(15)s also in the harder case where the two systems have the same underlying Steiner triple system. In the special case where the common STS is #19, an alternative method is given by means of the 1-factorizations of the complete graph K_8 associated to the two KTSs. Moreover, we present three new visual solutions to the schoolgirl problem, and we catalogue most of the classical (or interesting) solutions in the literature in terms of what KTS(15)s they are isomorphic to. This paper provides background on a classical topic, while shedding new light on the problem as well
Combinatorics, Probability and Computing
The main theme of this workshop was the use of probabilistic
methods in combinatorics and theoretical computer science. Although
these methods have been around for decades, they are being refined all
the time: they are getting more and more sophisticated and powerful.
Another theme was the study of random combinatorial structures,
either for their own sake, or to tackle extremal questions. The workshop
also emphasized connections between probabilistic combinatorics and
discrete probability
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