For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work [4] that no set of n parallel lines supports all n-vertex planar graphs. We show that intersecting lines, even if they ...

Let $n > k 1$ be integers, $[n] = \{1, \ldots, n\}$. $\mathcal F$ a family of $k$-subsets $[n]$. The is called intersecting if $F \cap F' \neq \varnothing$ for all $F, \in \mathcal F$. It almost it not but to every there at most one $F'\in satisfying \varnothing$. Gerbner et al. proved that \geq 2k + 2$ then $|\mathcal F| \leqslant {n - 1\choose 1}$ holds families. Our main result implies th...

2 Exercise. Prove it. ◦ In fact, as sophisticates can show you, on a complex space a linear operator which preserves all norms will also preserve all inner products. But the key thing is to remember is that a unitary operator preserves all inner products (and therefore all norms). • This is analogous to the fact that translations and rotations in ordinary three-dimensional space preserve distan...

At the free-fermion point, the six-vertex model with domain wall boundary conditions (DWBC) can be related to the Aztec diamond, a domino tiling problem. We study the mapping on the level of complete statistics for general domains and boundary conditions. This is obtained by associating to both models a set of non-intersecting lines in the Lindström-Gessel-Viennot (LGV) scheme. One of the conse...

We present some phenomenology of a new class of intersecting D-brane models. Soft SUSY breaking terms for these models are calculated in the complex structure (u)-moduli dominant SUSY breaking approach (in type IIA). In this case, the dependence of the soft terms on the Yukawas and Wilson lines drops out. These soft terms have a different pattern compared to the usual heterotic string models. P...

The phase diagram and the critical behaviour of an (intersecting) random surface gas model in three dimensions is studied by means of Monte Carlo simulations. The critical exponents (I, p, y and 8 are evaluated in the ‘critical window’ between the finite-size rounding and the correction-to-scaling regime. Within error bars, Ising exponents for the self-avoiding case (and along critical lines) a...

We study properties of the supersolid phase observed for hard-core bosons on the triangular lattice near half-integer filling factor, and the phase diagram of the system at finite temperature. We find that the solid order is always of the (2m, -m, -m) with m changing discontinuously from positive to negative values at half filling, in contrast with phases observed for Ising spins in a transvers...