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Abstract. We show that for a large class of cubic polynomials f , every sufficiently large number can be written as a sum of seven positive values of f . As a special case, we show that every number greater than e , reducing an open problem due to Pollock to a finite computation.

We study the following problem: given n real arguments a 1 , ..., a n and n real weights w 1 , ..., w n , under what conditions does the inequalityhold for all functions f satisfying f (k) ≥ 0 for some given integer k? Using simple combinatorial techniques, we can prove many generalizations of theorems ranging from the Fuchs inequality to the criterion for Schur convexity.

We consider the space [0, n] 3 , imagined as a three dimensional, axis-aligned grid world partitioned into n 3 1 × 1 × 1 unit cubes. Each cube is either considered to be empty, in which case a line of sight can pass through it, or obstructing, in which case no line of sight can pass through it. From a given position, some of these obstructing cubes block one's view of other obstructing cubes, leading to the following extremal problem: What is the largest number of obstructing cubes that can be simultaneously visible from the surface of an observer cube, over all possible choices of which cubes of [0, n] 3 are obstructing? We construct an example of a configuration in which Ω n 8 3 obstructing cubes are visible, and generalize this to an example with Ω n d− 1 d visible obstructing hypercubes for dimension d > 3. Using Fourier analytic techniques, we prove an O n d− 1 d log n upper bound in a reduced visibility setting.

Imagine that you are handed a rule for determining whether a cycle in a digraph is "good" or "bad", based on which edges of the cycle are traversed in the forward direction and which edges are traversed in the backward direction. Can you then construct a digraph which avoids having any "bad" cycles, but has arbitrarily large chromatic number?We answer this question when the rule is described in terms of a finite state machine. The proof relies on Nešetřil and Rödl's structural Ramsey theory of posets with a linear extension. As an application, we give a new proof of the Loop Lemma of Barto, Kozik, and Niven in the special case of bounded width algebras.