We investigate when limits of graphs (graphons) and permutations (permutons) are uniquely determined by finitely many densities of their substructures, i.e., when they are finitely forcible. Every permuton can be associated with a graphon through the notion of permutation graphs. We find permutons that are finitely forcible but the associated graphons are not. We also show that all permutons that can be expressed as a finite combination of monotone permutons and quasirandom permutons are finitely forcible, which is the permuton counterpart of the result of Lovász and Sós for graphons.

Graphons are analytic objects associated with convergent sequences of graphs. Problems from extremal combinatorics and theoretical computer science led to a study of graphons determined by finitely many subgraph densities, which are referred to as finitely forcible. Following the intuition that such graphons should have finitary structure, Lovász and Szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon is always compact. We disprove the conjecture by constructing a finitely forcible graphon such that the associated space is not compact. The construction method gives a general framework for constructing finitely forcible graphons with non-trivial properties.

Abstract. We prove that the number of Hamilton cycles in the random graph. Furthermore, we prove the hitting-time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree 2 creates ln n e n (1+o(1)) n Hamilton cycles a.a.s.

Abstract. In this paper we study the behaviour of the domination number of the Erdős-Rényi random graph G(n, p). Extending a result of Wieland and Godbole we show that the domination number of G(n, p) is equal to one of two values asymptotically almost surely whenever p ≫The explicit values are exactly at the first moment threshold, that is where the expected number of dominating sets starts to tend to infinity. For small p we also provide various non-concentration results which indicate why some sort of lower bound on the probability p is necessary in our first theorem. Concentration, though not on a constant length interval, is proven for every p ≫ 1/n. These results show that unlike in the case ofwhere concentration of the domination number happens around the first moment threshold, for p = O(ln n/n) it does so around the median. In particular, in this range the two are far apart from each other.

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