This internship is linked to the research project "Ordered structures: A meeting point between vector lattices, Banach spaces, graphs and metric geometry", at Instituto de Ciencias Matemáticas (www.icmat.es). The activities to be developed consist mainly of providing a smooth introduction to open problems at the frontier of knowledge of the discipline. The aim of this project is to foster an interdisciplinary approach to questions from Graph Theory, Metric Geometry and Functional Analysis. The student will have the opportunity to learn and collaborate with the supervisor and his collaborators. Specific tasks will be assigned depending on the background and experience at the Master or PhD levels.