Simplicial homeomorphs and trace-bounded hypergraphs

Our first main result is a uniform bound, in every dimension $k \in \mathbb N$, on the topological Tur\'an numbers of $k$-dimensional simplicial complexes: for each $k \in \mathbb N$, there is a $\lambda_k \ge k^{-2k^2}$ such that for any $k$-complex $\mathcal{S}$, every $k$-complex on $n \ge n_0(\mathcal{S})$ vertices with at least $n^{k+1 - \lambda_k}$ facets contains a homeomorphic copy of $\mathcal{S}$. This was previously known only in dimensions one and two, both by highly dimension-specific arguments: the existence of $\lambda_1$ is a result of Mader from 1967, and the existence of $\lambda_2$ was suggested by Linial in 2006 and recently proved by Keevash-Long-Narayanan-Scott. We deduce this geometric fact from a purely combinatorial result about trace-bounded hypergraphs, where an $r$-partite $r$-graph $H$ with partite classes $V_1, V_2, \dots, V_r$ is said to be $d$-trace-bounded if for each $2 \le i \le r$, all the vertices of $V_i$ have degree at most $d$ in the trace of $H$ on $V_1 \cup V_2 \cup \dots \cup V_i$. Our second main result is the following estimate for the Tur\'an numbers of degenerate trace-bounded hypergraphs: for all $r \ge 2$ and $d\in\mathbb N$, there is an $\alpha_{r,d} \ge (5rd)^{1-r}$ such that for any $d$-trace-bounded $r$-partite $r$-graph $H$, every $r$-graph on $n \ge n_0(H)$ vertices with at least $n^{r - \alpha_{r,d}}$ edges contains a copy of $H$. This strengthens a result of Conlon-Fox-Sudakov from 2009 who showed that such a bound holds for $r$-partite $r$-graphs $H$ satisfying the stronger hypothesis that the vertex-degrees in all but one of its partite classes are bounded (in $H$, as opposed to in its traces).Comment: 9 page

Bounding mean orders of sub-kk-trees of kk-trees

For a $k$-tree $T$, we prove that the maximum local mean order is attained in a $k$-clique of degree $1$ and that it is not more than twice the global mean order. We also bound the global mean order if $T$ has no $k$-cliques of degree $2$ and prove that for large order, the $k$-star attains the minimum global mean order. These results solve the remaining problems of Stephens and Oellermann [J. Graph Theory 88 (2018), 61-79] concerning the mean order of sub-$k$-trees of $k$-trees.Comment: 20 Pages, 6 Figure

Trees maximizing the number of almost-perfect matchings

We characterize the extremal trees that maximize the number of almost-perfect matchings, which are matchings covering all but one or two vertices, and those that maximize the number of strong almost-perfect matchings, which are matchings missing only one or two leaves. We also determine the trees that minimize the number of maximal matchings. We apply these results to extremal problems on the weighted Hosoya index for several choices of vertex-degree-based weight function.Comment: 21 pages, 8 figure

Algorithms for the ferromagnetic Potts model on expanders

We give algorithms for approximating the partition function of the ferromagnetic Potts model on $d$-regular expanding graphs. We require much weaker expansion than in previous works; for example, the expansion exhibited by the hypercube suffices. The main improvements come from a significantly sharper analysis of standard polymer models, using extremal graph theory and applications of Karger's algorithm to counting cuts that may be of independent interest. It is #BIS-hard to approximate the partition function at low temperatures on bounded-degree graphs, so our algorithm can be seen as evidence that hard instances of #BIS are rare. We believe that these methods can shed more light on other important problems such as sub-exponential algorithms for approximate counting problems.Comment: 27 page

New Stepping-Up Constructions for Multicoloured Hypergraphs

Generalizing the classical Ramsey numbers, $r_k(t; q, p)$ is the smallest integer $n$ such that every $q$-colouring of the $k$-sets on $n$ vertices contains a set of $t$ vertices spanning fewer than $p$ colours. We prove the first tower-type lower bounds on these numbers via two new stepping-up constructions, both variants of the original stepping-up lemma due to Erd\H{o}s and Hajnal. We use these to resolve a problem of Conlon, Fox, and R\"{o}dl. More precisely, we construct a family of hypergraphs with arbitrarily large tower height separation between their $2$-colour and $q$-colour Ramsey numbers.Comment: 15 page

Bounding Mean Orders of Sub-k-Trees of k-Trees

For a k-tree T, we prove that the maximum local mean order is attained in a k-clique of degree 1 and that it is not more than twice the global mean order. We also bound the global mean order if T has no k-cliques of degree 2 and prove that for large order, the k -star attains the minimum global mean order. These results solve the remaining problems of Stephens and Oellermann [J. Graph Theory 88 (2018), 61-79] concerning the mean order of sub-k-trees of k-trees

Plastic and natural inorganic microparticles do not differ in their effects on adult mussels (Mytilidae) from different geographic regions

Highlights: • First study to compare microplastic effects over a wide biogeographical range • Comparison between natural inorganic microparticles and plastic microparticles • Significant effects on byssus production, respiration and clearance rates, but small effect sizes • No ecologically relevant difference between impact of plastic and natural inorganic microparticles on Mytilidae Abstract: Microplastics are ubiquitous in the marine environment and studies on their effects on benthic filter feeders at least partly revealed a negative influence. However, it is still unclear whether the effects of microplastics differ from those of natural suspended microparticles, which constitute a common stressor in many coastal environments. We present a series of experiments that compared the effects of six-week exposures of marine mussels to two types of natural particles (red clay and diatom shells) to two types of plastic particles (Polymethyl Methacrylate and Polyvinyl Chloride). Mussels of the family Mytilidae from temperate regions (Japan, Chile, Tasmania) through subtropical (Israel) to tropical environments (Cabo Verde) were exposed to concentrations of 1.5 mg/L, 15 mg/L and 150 mg/L of the respective microparticles. At the end of this period, we found significant effects of suspended particles on respiration rate, byssus production and condition index of the animals. There was no significant effect on clearance rate and survival. Surprisingly, we observed only small differences between the effects of the different types of particles, which suggests that the mussels were generally equally robust towards exposure to variable concentrations of suspended solids regardless of whether they were natural or plastic. We conclude, that microplastics and suspended solids elicit similar effects on the tested response variables, and that both types of microparticles mainly cause acute responses rather than more persistent carry-over effects