### Discrete and Algorithmic Mathematics Red de Matemática Discreta y Algorítmica

# Recent trends

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*If you are interested in posting a short announcement, please contact Diego Ruano*.

### Recent trends V: The Weighted Region Problem

24 Jun 2024 Shortest path problems, where the goal is to find an optimal path between two points in the plane, are among the fundamental problems in computational geometry. These problems are crucial for a variety of applications, GPS navigation is perhaps the best-known application. However, there are numerous daily-life problems that can be solved using shortest path algorithms. (read more)

### Recent trends IV: From a naive question to a long-standing conjecture

04 May 2024 My aim is to talk about a question on numerical semigroups posed by Herbert Wilf (Wilf 1978) in the late seventies. This question deserves to be included in the gallery of easy-to-state but dangerous problems. That it is easy to state will soon become apparent. (read more)

### Recent trends III: Subsets of the integers without three term arithmetic progressions

22 Apr 2024 Let $A\subset \lbrace 1,\ldots, N\rbrace $ be a subset without non-trivial 3-term arithmetic progressions (APs), i.e., such that the equation $a+b=2c$ only has solutions for $a,b,c\in A$ if $a=b=c$. What is the maximum size $r_3(N):=\max_{A\subset \lbrace 1,\ldots,N\rbrace \text{ without 3 APs}}(|A|)$ that we can have? This question and many similar ones have driven the development of an important part of additive combinatorics during the last century (read more)

### Recent trends II: Codes, finite fields and skew polynomials

23 Mar 2024 Polynomials are used in virtually every area of Mathematics, and Coding Theory is not an exception. One of the most used error-correcting codes in practice are Reed-Solomon codes, which are defined as subspaces whose vectors are evaluations of polynomials (over a finite field $\mathbb{F}_q$) up to a certain degree (read more)

### Recent trends I: Geometry, randomness and Ramsey theory

23 Feb 2024 Since its inception in the early 1900s, Ramsey theory has flourished and become a cornerstone of modern discrete mathematics (and beyond). The central question asks to determine $r(s,t)$; the minimum integer $n\in \mathbb{N}$ such that any red/blue-colouring of the edges of the complete graph $K_n$ results in either a red $K_s$ or a blue $K_t$ (read more)