Golomb and Taylor (joined later by Etzion) have modified the notion of a complete Latin square to that of a Tuscan-\(k\) square. A Tuscan-\(k\) square is a row Latin square with the further property that for any two symbols \(a\) and \(b\) of the square, and for each \(m\) from \(1\) to \(k\), there is at most one row in which \(b\) is the \(m^{th}\) symbol to the right of \(a\). One question unresolved by a series of papers of the authors mentioned was whether or not \(n \times n\) Tuscan-\(2\) squares exist for infinitely many composite values of \(n+1\). It is shown here that if \(p\) is a prime and \(p \equiv 7 \pmod{12}\) or \(p \equiv 5 \pmod{24}\), then Tuscan-\(2\) squares of side \(2p\) exist. If \(p \equiv 7 \pmod{12}\), clearly \(2p + 1\) is always composite and if \(p \equiv 5 \pmod{24}\), \(2p+1\) is composite infinitely often. The squares constructed are in fact Latin squares that have the Tuscan-\(2\) property in both dimensions.

It is shown that if \([n] = X_1 \cup X_2 \cup \cdots \cup X_l\) is a partition of \([n]\) and if \(S_t\) is a family of \(t\)-valued functions intersecting on at least one element of \(k\) (circularly) consecutive blocks, then \(|S_t| < t^{n-k}\). If given \(a_1 < a_2 < \cdots < a_y \leq l \), \(\acute{S}_t\) is a family of \(t\)-valued functions intersecting on at least one element of \(X_{a_{1}+m}, X_{a_{2}+m}, \ldots, X_{a_{k}+m}\) for some \(m\) with \(1-a_1 \leq m \leq n – a_k\), then \(|\acute{S}_t| \leq t^{n-k}\). Both these results were conjectured by Faudree, Schelp, and Sós [FSS]. The main idea of our proofs is that of anticlusters introduced by Griggs and Walker [GW] which we discuss in some detail. We also discuss several related intersection theorems about sets, \(2\)-valued functions, and \(t\)-valued functions.

Let the edges of the complete graph \(K_n\) be \(2\)-colored. A Simple Hamiltonian Cycle is a Hamiltonian cycle in \(K_n\) that is either monochromatic or is a union of two monochromatic paths. The main result of this paper is that if \(n\) is an even integer greater than \(4\), then for every \(2\)-coloring of the edges of \(K_n\), there is a Simple Hamiltonian Cycle in \(K_n\) which is either monochromatic, or is a union of two monochromatic paths, where each path is of even length.

Uniquely pseudointersectable graphs are defined; this is closely related to the uniquely intersectable graphs introduced by Alter and Wang [1]. The S-property is necessary but not sufficient for a graph to be uniquely pseudointersectable. This condition is also sufficient for graphs with unique minimum cover. Finally, we show that for supercompact graphs, unique pseudointersectability and unique intersectability are equivalent. Thus we generalize some of the results in [1] to a wider class of graphs.