Consider two independent Erdős–Rényi G(N,1/2) graphs. We show that with probability tending to 1 as N→∞, the largest induced isomorphic subgraph has size either ⌊xN−εN⌋ or ⌊xN+εN⌋, where xN=4log2⁡N−2log2⁡log2⁡N−2log2⁡(4/e)+1 and εN=(4log2⁡N)−1/2. Using similar techniques, we also if Γ1 Γ2 are G(n,1/2) random graphs, then contains an copy of high n≤⌊yN−εN⌋ does not contain n>⌊yN+εN⌋, yN=2log2⁡N+...

We exhibit a complete set of identities for CNOT, the symmetric monoidal category generated by the controlled-not gate, the swap gate, and the computational ancillæ. We prove that CNOT is a discrete inverse category. Moreover, we prove that CNOT is equivalent to the category of partial isomorphisms of finitely-generated non-empty commutative torsors of characteristic 2. Equivalently this is the...

We establish a close connection between a reversible programming language based on type isomorphisms and a formally presented univalent universe. The correspondence relates combinators witnessing type isomorphisms in the programming language to paths in the univalent universe; and combinator optimizations in the programming language to 2-paths in the univalent universe. The result suggests a si...