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.