The random interlacement point process (introduced in [47], generalized in [ 50 ]) is a Poisson point process on the space of labeled doubly infinite nearest neighbour trajectories modulo time-shift on a transient graph G. We show that the random interlacement point process on any transient transitive graph G is a factor of i.i.d., i.e., it can be constructed from a family of i.i.d. random variables indexed by vertices of the graph via an equivariant measurable map.

The pandemic reminded us that the pathogen evolution still has a serious effect on human societies. States, however, can prepare themselves for the emergence of a novel pathogen with unknown characteristics by analysing potential scenarios. Game theory offers such an appropriate tool.

Goodman proved that the sum of the number of trianglesin a graph onnnodes and its complement is at least n3 / 24; in other words, this sum is minimized, asymptotically, by a random graph with edge density 1/2. Erdős conjectured that a similar inequality will hold for K4 in place of K3, but this was disproved by Thomason. But ananalogous statement does hold for some other graphs,which are called common graphs. Characterization ofcommon graphs seems, however, out of reach.

We introduce a new correlated percolation model on the d-dimensional lattice Zd called the random length worms model. Assume given a probability distribution on the set of positive integers (the length distribution) and v ∈ (0, ∞) (the intensity parameter). From each site of Zd we start POI(v) independent simple random walks with this length distribution. We investigate the connectivity properties of the set Sv of sites visited by this cloud of random walks.

A necklace can be considered as a cyclic list of n red and n blue beads in an arbitrary order. In the necklace folding problem the goal is to find a large crossing-free matching of pairs of beads of different colors in such a way that there exists a “folding” of the necklace, that is a partition into two contiguous arcs, which splits the beads of any matching edge into different arcs.

We present a construction that allows us to define a limit object of Banach space decorated graph sequences in a generalized homomorphism density sense. This general functional analytic framework provides a universal language for various combinatorial limit notions.

Our paper explores the game theoretic value of the 7-in-a-row game. We reduce the problem to solving a finite board game, which we target using Proof Number Search. We present a number of heuristic improvements to Proof Number Search and examine their effect within the context of this particular game. Although our paper does not solve the 7-in-a-row game, our experiments indicate that we have made significant progress towards it.