Let $\mathbf{x} = (x_1,\ldots,x_n)$, and $P(\mathbf{x}) \in \mathbb{Z}[\mathbf{x}]$. A Diophantine equation $P(\mathbf{x}) = 0$ is partition regular if, for any $r$-colouring $c: \mathbb{N} \to r$, there exists $c$-monochromatic $\mathbf{m} = (m_1,\ldots,m_n) \in \mathbb{N}^n$ such that $P(\mathbf{m}) = 0$. ...

Isaac Newton in his Principia (first book Ch. 12, Theorem XXXI) asserts that spherical shells exert no gravitational force in the cavity of the shell. This result was later extended to (ellipsoidal) homoeoid by P.-S. Laplace, using computation, and soon after by J. Ivory, using a more geometric approach; a homoeoid is the domain bounded by two homothetic ellipsoids with a common center. ...

Most of the above questions and open problems can naturally be stated for unbounded domains and half-spaces. Several authors have considered the scalar case of such problems and the literature is vast. The reader may consult paper [8], and the references therein for these problems. ...

In scalar case, the Bernoulli problem discussed earlier can also be obtained as a limit problem of the singular perturbation; literature are vast for this problem, so none mentioned none forgotten. A similar theory, yet to be developed, seems plausible for the system case, which seems to have much more possibilities and variations than its scalar counterpart. ...

Contrary to convexity problems, the symmetry methods, such as moving plane technique works very well for free boundary value problems for systems, when there is a symmetry in the given equation, the bulk domain, and boundary values. Here, we present the most simple example, leaving several obvious generalizations towards other problems to the reader. The original Saint Venant problem, with an overdetermination of the boundary gradient condition (here expressed as system) is to show that whenever there is a solution vector $\mathbf{u}$ to the following problem \begin{equation}\tag{1} \begin{cases} \Delta \mathbf{u} = - \mathbf{k} & \text{in } \Omega , \\ \mathbf{u} = \mathbf{0} & \text{on } \partial\Omega , \\ |\nabla \mathbf{u}| = 1 & \text{on } \partial\Omega , \\ \end{cases} \end{equation} the domain $\Omega $ has to be a ball. For scalar case, James Serrin [5] gave a very nice proof of this based on moving plane technique (of A. D. Alexandrov). ...

The system case of the free boundary for general $F(x, \mathbf{p})$, even if $F$ is independent of $x$, and convex in $\mathbf{p}$-variables, is of course a non-trivial problem. Let us consider the particular case as in ...

Convexity results for the Bernoulli type problems, represented in the system of equations \begin{equation}\begin{cases} \Delta \mathbf{u} = 0 & \text{in } \Omega \setminus \mathbf{D}, \\ |\nabla \mathbf{u}| = g(x) & \text{on } \partial\Omega . \\ \end{cases} \end{equation} seems for now out of reach, at least with the methods we know of. It is also not straightforward what conditions we should impose on the domains $\mathbf{D}$ (besides being convex and maybe $D_i$'s being homothetic). ...