We show that given $m$ disjoint chains in the Boolean lattice, we can create $m$ disjoint skipless chains that cover the same elements (where we call a chain skipless if any two consecutive elements differ in size by exactly one). By using this result we are able to answer two conjectures about the asymptotics of induced saturation numbers for the antichain, which are defined as follows. For positive integers $k$ and $n$, a family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ is $k$-antichain saturated if it does not contain an antichain of size $k$ (as induced subposet), but adding any set to $\mathcal{F}$ creates an antichain of size $k$. We use $\textrm{sat}^{\ast}(n,k)$ to denote the smallest size of such a family. With more work we pinpoint the exact value of $\textrm{sat}^{\ast}(n,k)$, for all $k$ and sufficiently large $n$. Previously, exact values for $\textrm{sat}^{\ast}(n,k)$ were only known for $k$ up to 6. We also show that for any poset $\mathcal{P}$, its induced saturation number (which is defined similar as for the antichain) grows at most polynomially: $\textrm{sat}^{\ast}(n, \mathcal{P})=O(n^c)$, where $c \leq |\mathcal{P}|^2/4+1$. This is based on joint works with Carla Groenland, Maria-Romina Ivan, Hugo Jacob and Tom Johnston.

Consider the following "controlled" random graph process: edges of the complete graph are revealed one by one in random order to an online algorithm, which immediately decides whether to retain each observed edge. The algorithm's objective is to construct a graph property within specified constraints on the total number of observed edges ("time") and the total number of retained edges ("budget").

During this talk, I will present results in this model for natural graph properties, such as connectivity, Hamiltonicity, and containment of fixed-size subgraphs. Specifically, I will describe a strategy to construct a Hamilton cycle at the hitting time for minimum degree 2 by retaining a linear number of edges. This extends the classical hitting time result for Hamiltonicity originally established by Ajtai–Komlós–Szemerédi and Bollobás.

The talk is based on joint work with Alan Frieze and Michael Krivelevich.