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If $k$ is a positive real number, we say that a set $S$ of real numbers is $k$-sum-free if there do not exist $x,y,z$ in $S$ such that $x + y = kz$. For $k$ greater than or equal to 4 we find the essentially unique measurable $k$-sum-free subset of $(0,1]$ of maximum size.

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Philip Hall's famous theorem on systems of distinct representatives and its not‐so‐famous improvement by Halmos and Vaughan (1950) can be regarded as statements about the existence of proper list‐colorings or list‐multicolorings of complete graphs. The necessary and sufficient condition for a proper “coloring” in these theorems has a rather natural generalization to a condition we call Hall's condition on a simple graph G, a vertex list assignment to G, and an assignment of nonnegative integers to the vertices of G. Hall's condition turns out to be necessary for the existence of a proper multicoloring of G under these assignments. The Hall‐Halmos‐Vaughan theorem may be stated: when G is a clique, Hall's condition is sufficient for the existence of a proper multicoloring. In this article, we undertake the study of the class HHV of simple graphs G for which Hall's condition is sufficient for the existence of a proper multicoloring. It is shown that HHV is contained in the class ℋ︁0 of graphs in which every block is a clique and each cut‐vertex lies in exactly two blocks. On the other hand, besides cliques, the only connected graphs we know to be in HHV are (i) any two cliques joined at a cut‐vertex, (ii) paths, and (iii) the two connected graphs of order 5 in ℋ︁0, which are neither cliques, paths, nor two cliques stuck together. In case (ii), we address the constructive aspect, the problem of deciding if there is a proper coloring and, if there is, of finding one. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 199–219, 2000

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Algebraic conditions and algorithmic procedures are given to determine whether an m x n rectangular configuration of switches can be transformed so that all switches are in the off position, regardless of initial configuration. However, when any switch is toggled, it and its rectilinearly adjacent neighbors change state. Using linear algebra, a k i t e field representation of the problem, and an analysis of Fibonacci polynomials, conditions on m and n are given which characterize when the m x n problem can be solved.

Given a subgraph G of the hypercube Q n , a coloring of the edges of Q n such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to G-polychromatically color the edges of any hypercube is called the polychromatic number of G. To determine polychromatic numbers, it is only necessary to consider a specific type of coloring, which we call simple. The main tool for finding upper bounds on polychromatic numbers is to translate the question of polychromatically coloring the hypercube so every embedding of a graph G contains every color into a question of coloring the 2-dimensional grid so that every so-called shape sequence corresponding to G contains every color. After surveying the tools for finding polychromatic numbers, we apply these techniques to find polychromatic numbers of a class of graphs called punctured hypercubes. We also consider the problem of finding polychromatic numbers in the setting where larger subcubes of the hypercube are colored. We exhibit two new constructions which show that this problem is not a straightforward generalization of the edge coloring problem.Recall that for an edge e ∈ E(Q n ), l(e) is the number of 1's to the left of the star in e, and r(e) is the number of 1's to the right. Call a coloring χ of the hypercube simple if χ(e) is determined by l(e) and r(e) (such colorings were called Ramsey in [6]). The following lemma tells us that when studying polychromatic colorings on the hypercube, we need only consider simple ones. The proof is essentially from [6], building on ideas from [1].Lemma 3 Let k ≥ 1 and G be a subgraph of Q k . If p(G) = r, then there is a simple G-polychromatic r-coloring on Q k .