Juanjo Rué

We prove that product-free sets of the free group over a finite alphabet have maximum density $1/2$ with respect to the natural measure that assigns total weight one to each set of irreducible words of a given size. This confirms a conjecture of Leader, Letzter, Narayanan and Walters. In more general terms, we actually prove that strongly $k$-product-free sets have maximum density $1/k$ in terms of the said measure.

We provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi and Bousquet-M\'elou [J. Comb. Theory Ser. B, 101 (2011), 315-377]. The decomposition of a map into 2- and 3-connected components allows us to obtain the generating functions of 2-and 3-connected bicoloured maps. Setting to zero the variable marking monochromatic edges we obtain the generating function of 3-connected bipartite maps, which is algebraic of degree 26. We deduce from it an asymptotic estimate for the number of 3-connected bipartite planar maps of the form $t \cdot n^{-5/2} \gamma^n$, where $\gamma=\rho^{-1} \approx 2.40958$ and $\rho \approx 0.41501$ is an algebraic number of degree 10.

We analyse uniform random cubic rooted planar maps and obtain limiting distributions for several parameters of interest.From the enumerative point of view, we present a unified approach for the enumeration of several classes of cubic planar maps, which allow us to recover known results in a more general and transparent way.This approach allows us to obtain new enumerative results. Concerning random maps, we first obtain the distribution of the degree of the root face, which has an exponential tail as for other classes of random maps. Our main result is a limiting map-Airy distribution law for the size of the largest block $L$, whose expectation is asymptotically $n/\sqrt{3}$ in a random cubic map with $n+2$ faces.We prove analogous results for the size of the largest cubic block, obtained from $L$ by erasing all vertices of degree two, and for the size of the largest 3-connected component, whose expected values are respectively $n/2$ and $n/4$.To obtain these results we need to analyse a new type of composition scheme which has not been treated by Banderier et al. [Random Structures Algorithms 2001].

We study an analogue of the Ramsey multiplicity problem for additive structures, establishing the minimum number of monochromatic $3$-APs in $3$-colorings of $\mathbb{F}_3^n$ and obtaining the first non-trivial lower bound for the minimum number of monochromatic $4$-APs in $2$-colorings of $\mathbb{F}_5^n$. The former parallels results by Cumings et al.~\cite{CummingsEtAl_2013} in extremal graph theory and the latter improves upon results of Saad and Wolf~\cite{SaadWolf_2017}. Lower bounds are notably obtained by extending the flag algebra calculus of Razborov~\cite{razborov2007flag}.

This work is a follow-up of the article [Proc.\ London Math.\ Soc.\ 119(2):358--378, 2019], where the authors solved the problem of counting labelled 4-regular planar graphs. In this paper, we obtain a precise asymptotic estimate for the number $g_n$ of labelled 4-regular planar graphs on $n$ vertices. Our estimate is of the form $g_n \sim g\cdot n^{-7/2} \rho^{-n} n!$, where $g>0$ is a constant and $\rho \approx 0.24377$ is the radius of convergence of the generating function $\sum_{n\ge 0}g_n x^n/n!$, and conforms to the universal pattern obtained previously in the enumeration of several classes of planar graphs. In addition to analytic methods, our solution needs intensive use of computer algebra in order to deal with large systems of multivariate polynomial equations. We also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular planar graphs, and for the number of 4-regular simple maps, both connected and 2-connected.

We analyse uniform random cubic rooted planar maps and obtain limiting distributions for several parameters of interest. From the enumerative point of view, we present a unified approach for the enumeration of several classes of cubic planar maps, which allow us to recover known results in a more general and transparent way. This approach allows us to obtain new enumerative results. Concerning random maps, we first obtain the distribution of the degree of the root face, which has an exponential tail as for other classes of random maps. Our main result is a limiting map-Airy distribution law for the size of the largest block $L$, whose expectation is asymptotically $n/\sqrt{3}$ in a random cubic map with $n+2$ faces. We prove analogous results for the size of the largest cubic block, obtained from $L$ by erasing all vertices of degree two, and for the size of the largest 3-connected component, whose expected values are respectively $n/2$ and $n/4$. To obtain these results we need to analyse a new type of composition scheme which has not been treated by Banderier et al. [Random Structures Algorithms 2001].

We consider the binomial random set model $[n]_p$ where each element in $\{1,\dots,n\}$ is chosen independently with probability $p:=p(n)$. We show that for essentially all regimes of $p$ and very general conditions for a matrix $A$ and a column vector $\mathbf{b}$, the count of specific integer solutions to the system of linear equations $A\mathbf{x} = \mathbf{b}$ with the entries of $\mathbf{x}$ in $[n]_p$ follows a (conveniently rescaled) normal limiting distribution. This applies among others to the number of solutions with every variable having a different value, as well as to a broader class of so-called non-trivial solutions in homogeneous strictly balanced systems. Our proof relies on the delicate linear algebraic study both of the subjacent matrices and the corresponding ranks of certain submatrices, together with the application of the method of moments in probability theory.