# Problems

Here are some open problems I like, with references to related papers of mine (not necessarily the origin of the problem).

- The
*(oriented) Ramsey number $r(H)$*of an (acyclic) directed graph $H$ is the smallest $N$ such that every tournament on $N$ vertices contains a copy of $H$. If $H$ has bounded degree, is $r(H)$ always bounded by a polynomial in the number of vertices of $H$? [reference] - A
*binary shuffle square of length $n$*is a bitstring $s\in {0,1}^n$ that can be partitioned into two disjoint identical subsequences. Thus, $101101$ and $110000$ are binary shuffle squares of length $6$, but $110010$ and $100111$ are not. If $n$ is even, is the number of binary shuffle squares of length $n$ asymptotic to $2^{n-1}$? [reference] - A permutation $\sigma$ is called
*$k$-universal*if it contains every permutation of length $k$ as a pattern. Is it true that a random permutation of length $1000k^2$ is $k$-universal with high probability? [reference] - Let $f(K_n, p)$ be the number of adjacency queries needed to find (with constant probability) a copy of a target graph $H$ in an infinite Erdős–Rényi random graph with edge probability $p$. It is known that as $p\rightarrow 0^+$, \(p^{-(2-\sqrt{2})n+O(1)} \ll f(K_n,p) \ll p^{-2n/3 +O(1)}\). What is the true growth rate of $f(K_n,p)$ as $p\rightarrow 0^+$? [reference]