Metric dimension is a graph parameter motivated by problems in robot navigation, drug design, and image processing. In this paper, we answer several open extremal problems on metric dimension and pattern avoidance in graphs from (Geneson, Metric dimension and pattern avoidance, Discrete Appl. Math. 284, 2020, 1-7). Specifically, we construct a new family of graphs that allows us to determine the maximum possible degree of a graph of metric dimension at most k, the maximum possible degeneracy of a graph of metric dimension at most k, the maximum possible chromatic number of a graph of metric dimension at most k, and the maximum n for which there exists a graph of metric dimension at most k that contains K n,n .We also investigate a variant of metric dimension called edge metric dimension and solve another problem from the same paper for n sufficiently large by showing that the edge metric dimension of P d n is d for n ≥ d d−1 . In addition, we use a probabilistic argument to make progress on another open problem from the same paper by showing that the maximum possible clique number of a graph of edge metric dimension at most k is 2 Θ(k) . We also make progress on a problem from (N. Zubrilina, On the edge dimension of a graph, Discrete Math. 341, 2018Math. 341, , 2083Math. 341, -2088 by finding a family of new triples (x, y, n) for which there exists a graph of metric dimension x, edge metric dimension y, and order n. In particular, we show that for each integer k > 0, there exist graphs G with metric dimension k, edge metric dimension 3 k (1 − o(1)), and order 3 k (1 + o(1)).

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The Ulam sequence is defined recursively as follows: a1 = 1, a2 = 2, and an, for n > 2, is the smallest integer not already in the sequence that can be written uniquely as the sum of two distinct earlier terms. This sequence is known for its mysterious quasi-periodic behavior and its surprising rigidity when we let a2 vary. This definition can be generalized to other sets of generators in different settings with a binary operation and a valid notion of size. Since there is not always a natural linear ordering of the elements, the resulting collections are called Ulam sets. Throughout the paper, we study Ulam sets in new settings. First, we investigate the structure of Ulam sets in noncommutative groups, in particular in free groups. We prove symmetry results, give conditions for certain words to be in the Ulam set, and prove a periodicity result for eventually periodic words with fixed prefixes. Then, we study Ulam sets in Z×(Z/nZ) and prove regularity for an infinite class of initial sets. We also examine an intriguing phenomenon about decompositions of later elements into sums of the generators. Finally, we consider a variant where we don't require the summands to be distinct, particularly in Z 2 .