Lukas Michel

The free semigroup $\mathcal{F}$ over a finite alphabet $\mathcal{A}$ is the set of all finite words with letters from $\mathcal{A}$ equipped with the operation of concatenation. A subset $S$ of $\mathcal{F}$ is $k$-product-free if no element of $S$ can be obtained by concatenating $k$ words from $S$, and strongly $k$-product-free if no element of $S$ is a (non-trivial) concatenation of at most $k$ words from $S$.

We prove that a $k$-product-free subset of $\mathcal{F}$ has upper Banach density at most $1/\rho(k)$, where $\rho(k) = \min\{\ell \colon \ell \nmid k - 1\}$. We also determine the structure of the extremal $k$-product-free subsets for all $k \notin \{3, 5, 7, 13\}$; a special case of this proves a conjecture of Leader, Letzter, Narayanan, and Walters. We further determine the structure of all strongly $k$-product-free sets with maximum density. Finally, we prove that $k$-product-free subsets of the free group have upper Banach density at most $1/\rho(k)$, which confirms a conjecture of Ortega, Rué, and Serra.

Colour the edges of the complete graph with vertex set $\{1, 2, \dotsc, n\}$ with an arbitrary number of colours. What is the smallest integer $f(\ell,k)$ such that if $n > f(\ell,k)$ then there must exist a monotone monochromatic path of length $\ell$ or a monotone rainbow path of length $k$? Lefmann, Rödl, and Thomas conjectured in 1992 that $f(\ell, k) = \ell^{k - 1}$ and proved this for $\ell \ge (3 k)^{2 k}$. We prove the conjecture for $\ell \geq k^3 (\log k)^{1 + o(1)}$ and establish the general upper bound $f(\ell, k) \leq k (\log k)^{1 + o(1)} \cdot \ell^{k - 1}$. This reduces the gap between the best lower and upper bounds from exponential to polynomial in $k$. We also generalise some of these results to the tournament setting.

Scott and Seymour conjectured the existence of a function $f \colon \mathbb{N} \to \mathbb{N}$ such that, for every graph $G$ and tournament $T$ on the same vertex set, $\chi(G)\ge f(k)$ implies that $\chi(G[N_T^+(v)])\ge k$ for some vertex $v$. In this note we disprove this conjecture even if $v$ is replaced by a vertex set of size $\mathcal{O}(\log{\lvert V(G) \rvert})$. As a consequence, we answer in the negative a question of Harutyunyan, Le, Thomassé, and Wu concerning the corresponding statement where the graph $G$ is replaced by another tournament, and disprove a related conjecture of Nguyen, Scott, and Seymour. We also show that the setting where chromatic number is replaced by degeneracy exhibits a quite different behaviour.

We prove that a large family of pairs of graphs satisfy a polynomial dependence in asymmetric graph removal lemmas. In particular, we give an unexpected answer to a question of Gishboliner, Shapira, and Wigderson by showing that for every $t \geq 4$, there are $K_t$-abundant graphs of chromatic number $t$. Using similar methods, we also extend work of Ruzsa by proving that a set $\mathcal{A} \subset \{1,\dots,N\}$ which avoids solutions with distinct integers to an equation of genus at least two has size $\mathcal{O}(\sqrt{N})$. The best previous bound was $N^{1 - o(1)}$ and the exponent of $1/2$ is best possible in such a result. Finally, we investigate the relationship between polynomial dependencies in asymmetric removal lemmas and the problem of avoiding integer solutions to equations. The results suggest a potentially deep correspondence. Many open questions remain.

We say that a family of permutations $t$-shatters a set if it induces at least $t$ distinct permutations on that set. What is the minimum number $f_k(n,t)$ of permutations of $\{1, \dots, n\}$ that $t$-shatter all subsets of size $k$? For $t \le 2$, $f_k(n,t) = \Theta(1)$. Spencer showed that $f_k(n,t) = \Theta(\log \log n)$ for $3 \le t \le k$ and $f_k(n,k!) = \Theta(\log n)$. In 1996, Füredi asked whether partial shattering with permutations must always fall into one of these three regimes. Johnson and Wickes recently settled the case $k = 3$ affirmatively and proved that $f_k(n,t) = \Theta(\log n)$ for $t > 2 (k-1)!$.

We give a surprising negative answer to the question of Füredi by showing that a fourth regime exists for $k \ge 4$. We establish that $f_k(n,t) = \Theta(\sqrt{\log n})$ for certain values of $t$ and prove that this is the only other regime when $k = 4$. We also show that $f_k(n,t) = \Theta(\log n)$ for $t > 2^{k-1}$. This greatly narrows the range of $t$ for which the asymptotic behaviour of $f_k(n,t)$ is unknown.