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We prove a known 2-coloring of the integers $[N] := \{1,2,3,\ldots,N\}$ minimizes the number of monochromatic arithmetic 3-progressions under certain restrictions. A monochromatic arithmetic progression is a set of equally-spaced integers that are all the same color. Previous work by Parrilo, Robertson and Saracino conjectured an optimal coloring for large $N$ that involves 12 colored blocks. Here, we prove that the conjecture is optimal among anti-symmetric colorings with 12 or fewer colored blocks. We leverage a connection to the coloring of the continuous interval $[0,1]$ used by Parrilo, Robertson, and Saracino as well as by Butler, Costello and Graham. Our proof identifies classes of colorings with permutations, then counts the permutations using mixed integer linear programming., Comment: 29 pages, 4 and 20 figures

Let $p_n$ be a random, degree $n$ polynomial whose roots are chosen independently according to the probability measure $\mu$ on the complex plane. For a deterministic point $\xi$ lying outside the support of $\mu$, we show that almost surely the polynomial $q_n(z):=p_n(z)(z - \xi)$ has a critical point at distance $O(1/n)$ from $\xi$. In other words, conditioning the random polynomials $p_n$ to have a root at $\xi$, almost surely forces a critical point near $\xi$. More generally, we prove an analogous result for the critical points of $q_n(z):=p_n(z)(z - \xi_1)\cdots (z - \xi_k)$, where $\xi_1, \ldots, \xi_k$ are deterministic. In addition, when $k=o(n)$, we show that the empirical distribution constructed from the critical points of $q_n$ converges to $\mu$ in probability as the degree tends to infinity, extending a recent result of Kabluchko., Comment: 39 pages, 5 figures; incorporated comments and suggestions from Boris Hanin

We study the problem of a policymaker who seeks to set policy optimally in an economy where the true economic structure is unobserved, and policymakers optimally learn from their observations of the economy. This is a classic problem of learning and control, variants of which have been studied in the past, but little with forward-looking variables which are a key component of modern policy-relevant models. As in most Bayesian learning problems, the optimal policy typically includes an experimentation component reflecting the endogeneity of information. We develop algorithms to solve numerically for the Bayesian optimal policy (BOP). However the BOP is only feasible in relatively small models, and thus we also consider a simpler specification we term adaptive optimal policy (AOP) which allows policymakers to update their beliefs but shortcuts the experimentation motive. In our setting, the AOP is significantly easier to compute, and in many cases provides a good approximation to the BOP. We provide a simple example to illustrate the role of learning and experimentation in an MJLQ framework.

We examine optimal and other monetary policies in a linear-quadratic setup with a relatively general form of model uncertainty, so-called Markov jump-linear-quadratic systems extended to include forward-looking variables and unobservable "modes." The form of model uncertainty our framework encompasses includes: simple i.i.d. model deviations; serially correlated model deviations; estimable regime-switching models; more complex structural uncertainty about very different models, for instance, backward- and forward-looking models; time-varying central-bank judgment about the state of model uncertainty; and so forth. We provide an algorithm for finding the optimal policy as well as solutions for arbitrary policy functions. This allows us to compute and plot consistent distribution forecasts - fan charts - of target variables and instruments. Our methods hence extend certainty equivalence and "mean forecast targeting" to more general certainty non-equivalence and "distribution forecast targeting."