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Hindman proved that, whenever the set $mathbb{N}$ of naturals is finitely colored, there must exist non-constant monochromatic solution of the equation $a+b=cd$. In this paper we extend this result for dense subsemigroups of $((0, infty), +)$ to near zero.
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We show that in any two-coloring of the positive integers there is a color for which the set of positive integers that can be represented as a sum of distinct elements with this color has upper logarithmic density at least $(2+sqrt{3})/4$ and this is best possible. This answers a forty-year-old question of ErdH{o}s.
Let $f(n,r)$ denote the maximum number of colourings of $A subseteq lbrace 1,ldots,nrbrace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $lbrace x,y,zrbrace$ such that $x+y=z$. We show that $f(n,2) = 2^{lceil n/2rceil}$, and describe the extremal subsets. Further, using linear optimisation, we asymptotically determine the logarithm of $f(n,r)$ for $r leq 5$. Similar results were obtained by H`an and Jimenez in the setting of finite abelian groups.
Given a graph $G$, one may ask: What sets of eigenvalues are possible over all weighted adjacency matrices of $G$? (The weight of an edge is positive or negative, while the diagonal entries can be any real numbers.) This is known as the Inverse Eigenvalue Problem for graphs (IEP-$G$). A mild relaxation of this question considers the multiplicity list instead of the exact eigenvalues themselves. That is, given a graph $G$ on $n$ vertices and an ordered partition $mathbf{m}= (m_1, ldots, m_ell)$ of $n$, is there a weighted adjacency matrix where the $i$-th distinct eigenvalue has multiplicity $m_i$? This is known as the ordered multiplicity IEP-$G$. Recent work solved the ordered multiplicity IEP-$G$ for all graphs on 6 vertices. In this work, we develop zero forcing methods for the ordered multiplicity IEP-$G$ in a multitude of different contexts. Namely, we utilize zero forcing parameters on powers of graphs to achieve bounds on consecutive multiplicities. We are able to provide general bounds on sums of multiplicities of eigenvalues for graphs. This includes new bounds on the the sums of multiplicities of consecutive eigenvalues as well as more specific bounds for trees. Using these results, we verify the previous results above regarding the IEP-$G$ on six vertices. In addition, applying our techniques to skew-symmetric matrices, we are able to determine all possible ordered multiplicity lists for skew-symmetric matrices for connected graphs on five vertices.
We give an abstract categorical treatment of Plonka sums and products using lax and oplax morphisms of monads. Plonka sums were originally defined as operations on algebras of regular theories. Their arities are sup-semilattices. It turns out that even more general operations are available on the categories of algebras of semi-analytic monads. Their arities are the categories of the regular polynomials over any sup-semilattice, i.e. any algebra for the terminal semi-analytic monad. We also show that similar operations can be defined on any category of algebras of any analytic monad. This time we can allow the arities to be the categories of linear polynomials over any commutative monoid, i.e. any algebra for the terminal analytic monad. There are also dual operations of Plonka products. They can be defined on Kleisli categories of commutative monads.
When the sequences of squares of primes is coloured with $K$ colours, where $K geq 1$ is an integer, let $s(K)$ be the smallest integer such that each sufficiently large integer can be written as a sum of no more than $s(K)$ squares of primes, all of the same colour. We show that $s(K) ll K expleft(frac{(3log 2 + {rm o}(1))log K}{log log K}right)$ for $K geq 2$. This improves on $s(K) ll_{epsilon} K^{2 +epsilon}$, which is the best available upper bound for $s(K)$.
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