Let $p_c$ and $q_c$ be the threshold and the expectation threshold, respectively, of an increasing family $F$ of subsets of a finite set $X$. Recently, Park and Pham proved Kahn–Kalai conjecture stating that a not-too-large multiple of $q_c$ is an upper bound on $p_c$. In the talk, I will present a slight improvement to the Park–Pham theorem, which is obtained from transferring the threshold result from the small $p$ regime to general $p$. Based on joint work with Oliver Riordan.

A factorisation $x=u_1u_2\cdots$ of an infinite word $x$ on alphabet $X$ is called ‘super-monochromatic’, for a given colouring of the finite words $X^{\ast}$ on alphabet $X$, if each word $u_{k_1}u_{k_2}\cdots u_{k_n}$, where $k_1<\cdots<k_n$, is the same colour. A direct application of Hindman’s theorem shows that if $x$ is eventually periodic, then for every finite colouring of $X^{\ast}$, there exist a suffix of $x$ that admits a super-monochromatic factorisation. What about the converse?

In this talk we show that the converse does indeed hold: thus a word $x$ is eventually periodic if and only if for every finite colouring of $X^{\ast}$ there is a suffix of $x$ having a super-monochromatic factorisation. This has been a conjecture in the community for some time. Our main tool is a Ramsey result about alternating sums. This provides a strong link between Ramsey theory and the combinatorics of infinite words.

Joint work with Imre Leader and Luca Q. Zamboni

Magnant and Martin conjectured that every $d$-regular graph on $n$ vertices can be covered by $n/(d+1)$ vertex-disjoint paths. Gruslys and Letzter verified this conjecture in the dense case, even for cycles rather than paths. We prove the analogous result for directed graphs and oriented graphs, that is, for all $\alpha>0$, there exists $n_0=n_0(\alpha)$ such that every $d$-regular digraph on $n$ vertices with $d \ge \alpha n $ can be covered by at most $n/(d+1)$ vertex-disjoint cycles. Moreover if $G$ is an oriented graph, then $n/(2d+1)$ cycles suffice. This also establishes Jackson's long standing conjecture for large $n$ that every $d$-regular oriented graph on $n$ vertices with $n\leq 4d+1$ is Hamiltonian.
This is joint work with Viresh Patel and  Mehmet Akif Yıldız.

Predictive scoring modelling is a common approach to measure financial health and credit worthiness in banking. Whilst the latter is a key factor in making decisions for lending, evaluating financial health helps to identify vulnerable customers trending towards financial hardship, who need support. The current macroeconomic uncertainties amplify the importance of extensive flexibility in modelling data solutions so that they can remain effective and adaptive to a volatile economic environment. This workshop is focused on discussing relevant techniques and mathematical methodologies which can help modernise traditional scoring models and accelerate innovation. In summary, the problem definition in the banking context is how automation and optimality can be achieved in a multiobjective problem where a subset of existing data features should be selected by relevance and uniqueness, assigned scoring weights by importance and how a pool of customers can be categorised accordingly using their individual scores and auto-adjusting thresholds of risk classification scales. The key challenge is imposed by the mutual dependency of the three sub-problems and their objectives. Introducing or removing constraints in any of them can change the feasibility and optimality of the others and the overall solution. It is common for traditional scoring models to be mainly focused on the predictive accuracy and their setup is often defined and revised manually, following ad-hoc exploratory data analysis and business-led decision making. An automated optimisation of the data features’ selection, scoring weights and classification thresholds definition can achieve respectively: ▪ Precise financial health evaluation and book classification under changing economic climate; ▪ Development of innovative data-driven solutions to enhance prevention from financial hardship and bankruptcies.