16 problems

For every finite tree $T$, define $i_k(T)=\left|\{I\subseteq V(T): I\text{ is independent and }|I|=k\}\right|$ and let $\alpha(T)$ be the maximum cardinality of an independent set…

For every integer $n\ge 0$ and every nonzero polynomial $p$ of degree $n$ whose zeros all lie on the unit circle $\mathbb{T}$, define $Q_n(z)=1+z^n$. For all $0\le s\le t\le\infty$…

The supplied source identifies Borcea's $2$-variance conjecture as the $p=2$ case of a conjecture concerning the distance between the critical points of a polynomial and those of i…

For every number of points $n\ge 1$, integers $\ell,k\ge 1$ with $k>\ell$, and configurations $x_1,\ldots,x_n\in\mathbb{R}^{\ell}$ and $y_1,\ldots,y_n\in\mathbb{R}^{k}$, define … I…

For every choice of positive masses $m_1,m_2,m_3,m_4>0$ and every prescribed cyclic ordering of four bodies, there exists exactly one strictly convex planar four-body central confi…

For the tetrahedron $K_4^{(3)}$, determine whether its Turán density satisfies $\pi(K_4^{(3)})=\frac{5}{9}$. Equivalently, if $\operatorname{ex}(n,K_4^{(3)})$ denotes the maximum n…

Determine exactly which prime numbers $p$ admit a simple rotationally symmetric Venn diagram consisting of $p$ Jordan curves on the sphere, invariant under a rotation through…

For every connected bipartite graph $G$ with $n$ vertices and every chosen vertex $v\in V(G)$, let $F_G$ be uniformly distributed over the finite set…

Let $n\ge 2$, let $(X,\mathcal{B},\mu)$ be a probability space, and let $T_1,\ldots,T_n$ be commuting measure-preserving transformations of $X$. For $f_1,\ldots,f_n\in L^\infty(X)$…

For every integer $n \ge 0$, every real number $K \ge 1$, and every nonempty set $A \subseteq \mathbb{F}_2^n$ satisfying $\lvert A+A\rvert \le K\lvert A\rvert$, there exist a subsp…

The supplied source identifies a source-corrected Written on the Wall II (WOWII) Graph Conjecture 160 and refers to an exact corrected statement, but does not provide that statemen…

The supplied source identifies this as a conjecture about finite graphs, with the relevant configuration involving a connected graph $G$ of radius $3$ having exactly one diametrica…

For every integer $k\ge 2$, the binomial-coefficient gcd quantity $D(k)$ specified in OEIS A080170 satisfies $D(k)=1$ if and only if…

Determine the normal orders of the arithmetic functions $h(n)$ and $H(n)$. The supplied source identifies this as Erdős Problem $\#696$, but does not define $h$, $H$, or specify th…

Determine whether there exists an integer $n>24$ such that $\max_{1\le m<n}\bigl(m+\tau(m)\bigr)\le n+2$, where $\tau(m)$ denotes the number of positive divisors of $m$. Equivalent…

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