Oriol Serra’s research while affiliated with Polytechnic University of Catalonia and other places

Given a hypergraph H=(V,E), define for every edge $e\in E$ a linear expression with arguments corresponding with the vertices. Next, let the polynomial $p_H$ be the product of such linear expressions for all edges. Our main goal was to find a relationship between the Alon-Tarsi number of $p_H$ and the edge density of H. We prove that $AT(p_H)=\lceil ed(H)\rceil+1$ if all the coefficients in $p_H$ are equal to 1. Our main result is that, no matter what those coefficients are, they can be permuted within the edges so that for the resulting polynomial $p_H^\prime$, $AT(p_H^\prime)\leq 2\lceil ed(H)\rceil+1$ holds. We conjecture that, in fact, permuting the coefficients is not necessary. If this were true, then in particular a significant generalization of the famous 1-2-3 Conjecture would follow.

The Lonely Runner Conjecture originated in Diophantine approximation will turn 60 in 2028. Even if the conjecture is still widely open, the flow of partial results, innovative tools and connections to different problems and applications has been steady on its long life. This survey attempts to give a panoramic view of the status of the problem, trying to highlight the contributions of the many papers that it has originated.

An edge-colored multigraph G is rainbow connected if every pair of vertices is joined by at least one rainbow path, i.e., a path where no two edges are of the same color. In the context of multilayered networks we introduce the notion of multilayered random geometric graphs, from $h\ge 2$ independent random geometric graphs G(n,r) on the unit square. We define an edge-coloring by coloring the edges according to the copy of G(n,r) they belong to and study the rainbow connectivity of the resulting edge-colored multigraph. We show that $r(n)=\left(\frac{\log n}{n}\right)^{\frac{h-1}{2h}}$ is a threshold of the radius for the property of being rainbow connected. This complements the known analogous results for the multilayerd graphs defined on the Erd\H{o}s-R\' enyi random model.

We prove that product‐free subsets of the free group over a finite alphabet have maximum upper density with respect to the natural measure that assigns total weight one to each set of irreducible words of a given length. This confirms a conjecture of Leader, Letzter, Narayanan, and Walters. In more general terms, we actually prove that strongly ‐product‐free sets have maximum upper density in terms of this measure. The bounds are tight.

We show that if A is a subset of a group of prime order p such that $|2A|<2.7652|A|$ and $|A|<1.25\cdot10^{-6}p$, then A is contained in an arithmetic progression with at most $|2A|-|A|+1$ terms, and 2A contains an arithmetic progression with the same difference and at least $2|A|-1$ terms. This improves a number of previously known results towards the conjectured value $3|A|-4$ for which such an statement should hold..

In \cite[Serra, Vena, Extremal families for the Kruskal-Katona theorem]{sv21}, the authors have shown a characterization of the extremal families for the Kruskal-Katona Theorem. We further develop some of the arguments given in \cite{sv21} and give additional properties of these extremal families. F\"uredi-Griggs/M\"ors theorem from 1986/85 \cite{furgri86,mors85} claims that, for some cardinalities, the initial segment of the colexicographical is the unique extremal family; we extend their result as follows: the number of (non-isomorphic) extremal families strictly grows with the gap between the last two coefficients of the k-binomial decomposition. We also show that every family is an induced subfamily of an extremal family, and that, somewhat going in the opposite direction, every extremal family is close to being the inital segment of the colex order; namely, if the family is extremal, then after performing t lower shadows, with $t=O(\log(\log n))$, we obtain the initial segment of the colexicographical order. We also give a ``fast'' algorithm to determine whether, for a given t and m, there exists an extremal family of size m for which its t-th lower shadow is not yet the initial segment in the colexicographical order. As a byproduct of these arguments, we give yet another characterization of the families of k-sets satisfying equality in the Kruskal--Katona theorem. Such characterization is, at first glance, less appealing than the one in \cite{sv21}, since the additional information that it provides is indirect. However, the arguments used to prove such characterization provide additional insight on the structure of the extremal families themselves.

Given a family S of k--subsets of [n], its lower shadow $\Delta(S)$ is the family of $(k-1)$--subsets which are contained in at least one set in S. The celebrated Kruskal--Katona theorem gives the minimum cardinality of $\Delta(S)$ in terms of the cardinality of S. F\"uredi and Griggs (and M\"ors) showed that the extremal families for this shadow minimization problem in the Boolean lattice are unique for some cardinalities and asked for a general characterization of these extremal families. In this paper we prove a new combinatorial inequality from which yet another simple proof of the Kruskal--Katona theorem can be derived. The inequality can be used to obtain a characterization of the extremal families for this minimization problem, giving an answer to the question of F\"uredi and Griggs. Some known and new additional properties of extremal families can also be easily derived from the inequality.