Matching measure, Benjamini-Schramm convergence and the monomer-dimer free energy

We define the matching measure of a lattice L as the spectral measure of the tree of self-avoiding walks in L. We connect this invariant to the monomer-dimer partition function of a sequence of finite graphs converging to L. This allows us to express the monomer-dimer free energy of L in terms of the measure. Exploiting an analytic advantage of the matching measure over the Mayer series then leads to new, rigorous bounds on the monomer-dimer free energies of various Euclidean lattices. While our estimates use only the computational data given in previous papers, they improve the known bounds significantly.Comment: 18 pages, 3 figure

Almost-monochromatic sets and the chromatic number of the plane

In a colouring of $\mathbb{R}^d$ a pair $(S,s_0)$ with $S\subseteq \mathbb{R}^d$ and with $s_0\in S$ is \emph{almost monochromatic} if $S\setminus \{s_0\}$ is monochromatic but $S$ is not. We consider questions about finding almost monochromatic similar copies of pairs $(S,s_0)$ in colourings of $\mathbb{R}^d$, $\mathbb{Z}^d$, and in $\mathbb{Q}$ under some restrictions on the colouring. Among other results, we characterise those $(S,s_0)$ with $S\subseteq \mathbb{Z}$ for which every finite colouring of $\mathbb{R}$ without an infinite monochromatic arithmetic progression contains an almost monochromatic similar copy of $(S,s_0)$. We also show that if $S\subseteq \mathbb{Z}^d$ and $s_0$ is outside of the convex hull of $S\setminus \{s_0\}$, then every finite colouring of $\mathbb{R}^d$ without a similar monochromatic copy of $\mathbb{Z}^d$ contains an almost monochromatic similar copy of $(S,s_0)$. Further, we propose an approach of finding almost-monochromatic sets that might lead to a non-computer assisted proof of $\chi(\R^2)\geq 5$

Almost-monochromatic sets and the chromatic number of the plane

In a colouring of Rd a pair (S, s0) with S ⊆ Rd and with s0 ∈ S is almost-monochromatic if S \ {s0} is monochromatic but S is not. We consider questions about finding almost-monochromatic similar copies of pairs (S, s0) in colourings of Rd, Zd, and of Q under some restrictions on the colouring. Among other results, we characterise those (S, s0) with S ⊆ Z for which every finite colouring of R without an infinite monochromatic arithmetic progression contains an almost-monochromatic similar copy of (S, s0). We also show that if S ⊆ Zd and s0 is outside of the convex hull of S \ {s0}, then every finite colouring of Rd without a monochromatic similar copy of Zd contains an almost-monochromatic similar copy of (S, s0). Further, we propose an approach based on finding almost-monochromatic sets that might lead to a human-verifiable proof of χ(R2) ≥ 5

Photosynthetic performance of two freshwater red algal species

Photosynthetic performances of two freshwater red algal populations from freshwaters of the Carpathian basin were measured in this study. Populations were collected from different habitats: Bangia atropurpurea from Lake Balaton and Batrachospermum gelatinosum from the Tapolca stream. Their photosynthesis was studied in a wide range of temperature (5–35 °C) and light intensity (0–1150 μmol m–2 s‑1) in a photosynthetron. We found both species’ photosynthesis maxima at 25°C but B. atropurpurea had significantly higher photosynthetic production. Low and medium values were calculated for the species’ photoadaptation parameters. Compensation light intensities determined in this study were similar to those obtained in previous studies. Both species utilized light efficiently; photoinhibition was detected only at two measuring temperatures for Bangia and at four measuring temperatures for Batrachospermum. P-T characteristics of the species revealed that both have temperature optima at 25 °C under high and medium light intensities but there are no such remarkable optima at low irradiance. The biomass specific respiration of both species increased with increasing temperature. We confirmed the good light utilization of these red algal species but found temperature optima higher than reported previously