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1. When Sets Can and Cannot Have MSTD Subsets

   Chu, Hung; McNew, Nathan; Miller, Steven J; Xu, Victor; Zhang, Sean (January 2018, Journal of integer sequences)

   A finite set of integers A is a sum-dominant (also called a More Sums Than Differences or MSTD) set if |A+A| > |A−A|. While almost all subsets of {0, . . . , n} are not sum-dominant, interestingly a small positive percentage are. We explore sufficient conditions on infinite sets of positive integers such that there are either no sum-dominant subsets, at most finitely many sum-dominant subsets, or infinitely many sum-dominant subsets. In particular, we prove no subset of the Fibonacci numbers is a sum-dominant set, establish conditions such that solutions to a recurrence relation have only finitely many sum-dominant subsets, and show there are infinitely many sum-dominant subsets of the primes. 

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2. Cup products in the etale cohomology of number fields

   F. M. Bleher, T. Chinburg (August 2018, New York journal of mathematics)

   This paper concerns cup product pairings in \'etale cohomology related to work of M. Kim and of W. McCallum and R. Sharifi. We will show that by considering Ext groups rather than cohomology groups, one arrives at a pairing which combines invariants defined by Kim with a pairing defined by McCallum and Sharifi. We also prove a formula for Kim's invariant in terms of Artin maps in the case of cyclic unramified Kummer extensions. One consequence is that for all integers $n>1$, there are infinitely many number fields over which there are both trivial and non-trivial Kim invariants associated to cyclic groups of order $$n$$. 

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3. An exceptional splitting of Khovanov’s arc algebras in characteristic 2

   https://doi.org/10.4064/fm230712-2-12  

   Cohen, Jesse (January 2024, Fundamenta Mathematicae)

   We show that there is an associative algebra $$\tilde{H}_n$$ such that, over a base ring R of characteristic 2, Khovanov's arc algebra $$H_n$$ is isomorphic to the algebra $$\tilde{H}_n[x]/(x^2)$$. We also show a similar result for bimodules associated to planar tangles and prove that there is no such isomorphism over the integers. 

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4. On Explicit Constructions of Designs

   https://doi.org/10.37236/10513  

   Liu, Xizhi; Mubayi, Dhruv (January 2022, The Electronic Journal of Combinatorics)

   An $(n,r,s)$-system is an $$r$$-uniform hypergraph on $$n$$ vertices such that every pair of edges has an intersection of size less than $$s$$. Using probabilistic arguments, Rödl and Šiňajová showed that for all fixed integers $$r> s \ge 2$$, there exists an $(n,r,s)$-system with independence number $$O\left(n^{1-\delta+o(1)}\right)$$ for some optimal constant $$\delta >0$$ only related to $$r$$ and $$s$$. We show that for certain pairs $(r,s)$ with $$s\le r/2$$ there exists an explicit construction of an $(n,r,s)$-system with independence number $$O\left(n^{1-\epsilon}\right)$$, where $$\epsilon > 0$$ is an absolute constant only related to $$r$$ and $$s$$. Previously this was known only for $s>r/2$ by results of Chattopadhyay and Goodman. 

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5. Contact surgery numbers

   https://doi.org/10.4310/JSG.2023.v21.n6.a4  

   Etnyre, John; Kegel, Marc; Onaran, Sinem (January 2023, Journal of Symplectic Geometry)

   It is known that any contact $$3$$-manifold can be obtained by rationally contact Dehn surgery along a Legendrian link $$L$$ in the standard tight contact $$3$$-sphere. We define and study various versions of contact surgery numbers, the minimal number of components of a surgery link $$L$$ describing a given contact $$3$$-manifold under consideration. In the first part of the paper, we relate contact surgery numbers to other invariants in terms of various inequalities. In particular, we show that the contact surgery number of a contact manifold is bounded from above by the topological surgery number of the underlying topological manifold plus three. In the second part, we compute contact surgery numbers of all contact structures on the $$3$$-sphere. Moreover, we completely classify the contact structures with contact surgery number one on $$S^1\times S^2$$, the Poincar\'e homology sphere and the Brieskorn sphere $$\Sigma(2,3,7)$$. We conclude that there exist infinitely many non-isotopic contact structures on each of the above manifolds which cannot be obtained by a single rational contact surgery from the standard tight contact $$3$$-sphere. We further obtain results for the $$3$$-torus and lens spaces. As one ingredient of the proofs of the above results we generalize computations of the homotopical invariants of contact structures to contact surgeries with more general surgery coefficients which might be of independent interest. 

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