London School of Economics

An interesting twist on classical subgraph containment problems in graph theory is the following: given a graph $H$ and a collection $\{G_1, \dots , G_m\}$ of graphs on a common vertex set $[n]$, what conditions on $G_i$ guarantee a copy of $H$ using at most one edge from each $G_i$? Such a subgraph is called transversal, and the above problem is closely related to the study of temporal graphs in Network Theory. In 2020 Joos and Kim showed that if $\delta(G_i)\geq n/2$, the collection contains a transversal Hamilton cycle. We improve on their result by showing that it actually contains every transversal Hamilton cycle if $\delta(G_i)\geq (1/2+o(1))n$. That is, for every function $\chi:[n]\to[m]$, there is a Hamilton cycle whose $i$-th edge belongs to $G_{\chi(i)}$.

This is joint work with Candida Bowtell, Patrick Morris and Katherine Staden.

A colouring rule is a way to colour the points $x$ of a probability space according to the colours of finitely many measure preserving tranformations of $x$. The rule is paradoxical if the rule can be satisfied a.e. by some colourings, but by none whose inverse images are measurable with respect to any finitely additive extension for which the transformations remain measure preserving. We show that proper graph colouring as a rule can be paradoxical. And we demonstrate rules defined via optimisation that are paradoxical. A connection to measure theoretic paradoxes is established.

Mean-field games with absorption is a class of games, that have been introduced in Campi and Fischer (2018) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this talk, we push the study of such games further, extending their scope along two main directions. First, a direct dependence on past absorptions has been introduced in the drift of players' state dynamics. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth. Therefore, the mean-field interaction among the players takes place in two ways: via the empirical sub-probability measure of the surviving players and through a process representing the fraction of past absorptions over time. Moreover, relaxing the boundedness of the coefficients allows for more realistic dynamics for players' private states. We prove existence of solutions of the mean-field game in strict as well as relaxed feedback form. Finally, we show that such solutions induce approximate Nash equilibria for the N-player game with vanishing error in the mean-field limit as N goes to infinity. This is based on a joint work with Maddalena Ghio and Giulia Livieri (SNS Pisa).