The First Classical Ramsey Number for Hypergraphs is Computed

With the help of the computer, we have shown that in any coloring with two colors of the triangles on a set of 13 points there must exist a monochromatic tetrahedron. This proves the new upper bound R (4,4;3) \u3c = 13. The previous best upper bound of 15 was derived independently by Giraud (1969 [2]), Schwenk (1978 [5]) and Sidorenko (1980 [6]). The first construction of a R (4,4;3)-good hypergraph on 12 points was presented by Isbell (1969 [3]), and the same one again more elegantly by Sidorenko (1980 [6]). We have constructed more than 200,000 R (4,4;3)-good hypergraphs on 12 points, but probably not the full set. R (4,4;3)=13 is the first known exact value of a classical Ramsey number for hypergraphs. The solution was achieved with the help of a variety of algorithms relying on a strong connection between the colorings with two colors of the triangles on n points and the so-called Tura´n set systems T(n ,5,4). The main criterion used to prune the search space for R (4,4;3)-good hypergraphs was to count the number of 4-sets containing two triangles of each color; such families of 4-sets are known to form Tura´n systems and their cardinalities must be minorized by the corresponding Tura´n numbers T(n ,5,4). We used an innovative method for generating large families of set systems which efficiently prevents isomorphic copies of set systems being produced. This method has many potential applications to other general computer searches for elusive combinatorial configurations. As a check on the correctness of the algorithms, many of the intermediate subfamilies of R (4,4;3)-good hypergraphs were generated by two different methods: from colorings of triangles on a smaller number of points and independently via Tura´n systems. An important component of the software used was a general set-system automorphism group program [4]

Two Generalizations of Homogeneity in Groups with Applications to Regular Semigroups

Let $X$ be a finite set such that $|X|=n$ and let $i\leq j \leq n$. A group G\leq \sym is said to be $(i,j)$-homogeneous if for every $I,J\subseteq X$, such that $|I|=i$ and $|J|=j$, there exists $g\in G$ such that $Ig\subseteq J$. (Clearly $(i,i)$-homogeneity is $i$-homogeneity in the usual sense.) A group G\leq \sym is said to have the $k$-universal transversal property if given any set $I\subseteq X$ (with $|I|=k$) and any partition $P$ of $X$ into $k$ blocks, there exists $g\in G$ such that $Ig$ is a section for $P$. (That is, the orbit of each $k$-subset of $X$ contains a section for each $k$-partition of $X$.) In this paper we classify the groups with the $k$-universal transversal property (with the exception of two classes of 2-homogeneous groups) and the $(k-1,k)$-homogeneous groups (for $2<k\leq \lfloor \frac{n+1}{2}\rfloor$). As a corollary of the classification we prove that a $(k-1,k)$-homogeneous group is also $(k-2,k-1)$-homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the $k$-universal transversal property have the $(k-1)$-universal transversal property. A corollary of all the previous results is a classification of the groups that together with any rank $k$ transformation on $X$ generate a regular semigroup (for $1\leq k\leq \lfloor \frac{n+1}{2}\rfloor$). The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.Comment: Includes changes suggested by the referee of the Transactions of the AMS. We gratefully thank the referee for an outstanding report that was very helpful. We also thank Peter M. Neumann for the enlightening conversations at the early stages of this investigatio

A Potpourri of Partition Properties

The cardinal characteristic inequality r <= hm3 is proved. Several partition relations for ordinals and one for countable scattered types are given. Moreover partition relations for lexicographically ordered sequences of zeros and ones are given in a no-choice context