Heterochromatic Higher Order Transversals for Convex Sets

In this short paper, we show that if $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be a collection of families compact $(r, R)$-fat convex sets in $\mathbb{R}^{d}$ and if every heterochromatic sequence with respect to $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ contains $k+2$ convex sets that can be pierced by a $k$-flat then there exists a family $\mathcal{F}_{m}$ from the collection that can be pierced by finitely many $k$-flats. Additionally, we show that if $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ be a collection of families of compact convex sets in $\mathbb{R}^{d}$ where each $\mathcal{F}_{n}$ is a family of closed balls (axis parallel boxes) in $\mathbb{R}^{d}$ and every heterochromatic sequence with respect to $\left\{\mathcal{F}_{n}\right\}_{n \in \mathbb{N}}$ contains $2$ intersecting closed balls (boxes) then there exists a family $\mathcal{F}_{m}$ from the collection that can be pierced by a finite number of points from $\mathbb{R}^{d}$. To complement the above results, we also establish some impossibility of proving similar results for other more general families of convex sets. Our results are a generalization of $(\aleph_0,k+2)$-Theorem for $k$-transversals of convex sets by Keller and Perles (Symposium on Computational Geometry 2022), and can also be seen as a colorful infinite variant of $(p,q)$-Theorems of Alon and Klietman (Advances in Mathematics 1992), and Alon and Kalai (Discrete & Computational Geometry 1995).Comment: 16 pages and 5 figures. Section 3 rewritte

Dimension Independent Helly Theorem for Lines and Flats

We give a generalization of dimension independent Helly Theorem of Adiprasito, B\'{a}r\'{a}ny, Mustafa, and Terpai (Discrete & Computational Geometry 2022) to higher dimensional transversal. We also prove some impossibility results that establish the tightness of our extension.Comment: 10 page

Stabbing boxes with finitely many axis-parallel lines and flats

We give necessary and sufficient condition for an infinite collection of axis-parallel boxes in $\mathbb{R}^{d}$ to be pierceable by finitely many axis-parallel $k$-flats, where $0 \leq k < d$. We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner $(p,q)$-problem.Comment: 13 page

Almost covering all the layers of hypercube with multiplicities

Given a hypercube $\mathcal{Q}^{n} := \{0,1\}^{n}$ in $\mathbb{R}^{n}$ and $k \in \{0, \dots, n\}$, the $k$-th layer $\mathcal{Q}^{n}_{k}$ of $\mathcal{Q}^{n}$ denotes the set of all points in $\mathcal{Q}^{n}$ whose coordinates contain exactly $k$ many ones. For a fixed $t \in \mathbb{N}$ and $k \in \{0, \dots, n\}$, let $P \in \mathbb{R}\left[x_{1}, \dots, x_{n}\right]$ be a polynomial that has zeroes of multiplicity at least $t$ at all points of $\mathcal{Q}^{n} \setminus \mathcal{Q}^{n}_{k}$, and $P$ has zeros of multiplicity exactly $t-1$ at all points of $\mathcal{Q}^{n}_{k}$. In this short note, we show that $$deg(P) \geq \max\left\{ k, n-k\right\}+2t-2.$$Matching the above lower bound we give an explicit construction of a family of hyperplanes $H_{1}, \dots, H_{m}$ in $\mathbb{R}^{n}$, where $m = \max\left\{ k, n-k\right\}+2t-2$, such that every point of $\mathcal{Q}^{n}_{k}$ will be covered exactly $t-1$ times, and every other point of $\mathcal{Q}^{n}$ will be covered at least $t$ times. Note that putting $k = 0$ and $t=1$, we recover the much celebrated covering result of Alon and F\"uredi (European Journal of Combinatorics, 1993). Using the above family of hyperplanes we disprove a conjecture of Venkitesh (The Electronic Journal of Combinatorics, 2022) on exactly covering symmetric subsets of hypercube $\mathcal{Q}^{n}$ with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest. We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.Comment: 16 pages, substantial changes from previous version, title and abstract changed to better reflect the content of the pape

On higher multiplicity hyperplane and polynomial covers for symmetry preserving subsets of the hypercube

Alon and F\"uredi (European J. Combin. 1993) gave a tight bound for the following hyperplane covering problem: find the minimum number of hyperplanes required to cover all points of the n-dimensional hypercube {0,1}^n except the origin. Their proof is among the early instances of the polynomial method, which considers a natural polynomial (a product of linear factors) associated to the hyperplane arrangement, and gives a lower bound on its degree, whilst being oblivious to the (product) structure of the polynomial. Thus, their proof gives a lower bound for a weaker polynomial covering problem, and it turns out that this bound is tight for the stronger hyperplane covering problem. In a similar vein, solutions to some other hyperplane covering problems were obtained, via solutions of corresponding weaker polynomial covering problems, in some special cases in the works of the fourth author (Electron. J. Combin. 2022), and the first three authors (Discrete Math. 2023). In this work, we build on these and solve a hyperplane covering problem for general symmetric sets of the hypercube, where we consider hyperplane covers with higher multiplicities. We see that even in this generality, it is enough to solve the corresponding polynomial covering problem. Further, this seems to be the limit of this approach as far as covering symmetry preserving subsets of the hypercube is concerned. We gather evidence for this by considering the class of blockwise symmetric sets of the hypercube (which is a strictly larger class than symmetric sets), and note that the same proof technique seems to only solve the polynomial covering problem

Nutrient conditions and chironomid productivity in Kolkata, India: assessment for biomonitoring and ecological management

The chironomid diversity in the water bodies are useful indicators of the nutrient and environmental states. A spatial scale analysis on the relative abundance of the chironomid species in the context of selected nutrient indicators like organic carbon (C), potassium ions (K+), nitrate (NO3¯), and phosphate (PO42¯) of the water bodies was assessed to justify the use of chironomids in environmental biomonitoring. Analysis of a sample of 90 data from eight different ponds of Kolkata, India, revealed the presence of 11 chironomid species in different relative densities. The chironomid immature productivity was found to be positively correlated with C and PO42¯ of the water bodies, while no definite significant correlation was observed for K+, NO3¯. Based on these nutrients and the productivity of chironomids the ponds could be distinguished from one another. The abundance of three species of chironomid midges, Chironomus striatipennis, Chironomus circumdatus and Kiefferulus calligaster were prominent in all the water bodies. Cluster analysis showed that these species were highly correlated in their abundance contrast to others. The correspondence analysis showed distribution of the chironomid species to differ against the variance of nutrients. The results are supportive of the use of chironomid larvae in biomonitoring and ecological restoration of urban water bodies, through monitoring the nutrient status and the chironomid species assemblage. In this instance the chironomid species C. striatipennis, C. circumdatus and K. calligaster can specifically act as indicator of the nutrient state of the ponds

Chironomid midges as allergens: evidence from two species from West Bengal, Kolkata, India

Background & objectives: Arthropods of different taxonomic identity including chironomid midges are known to induce allergic response in humans. The present study was done to access two common chironomid species Chironomus circumdatus and Polypedilum nubifer for their sensitizing potential as an allergen in atopic patients and controls. Methods: Following preparation of allergenic extracts of the two chironomid species separately, 198 atopic patients attending an allergy clinic and 50 age matched controls were tested along with a routine panel of allergens to assess sensitization. Results: The skin prick test (SPT) results revealed that 189 of the 198 patients (95.4%) demonstrated sensitization to both the chironomid species. Higher levels of total IgE was observed in atopic subjects than in the control group. Interpretation & conclusions: The results suggest that the chironomid midges Chironomus circumdatus and Polypedilum nubifer can elicit sensitization in humans. A potential risk for allergic reactions by susceptible individuals exists due to these chironomid species, owing to their abundance and chances of contact with human beings. Further studies may be initiated to characterize the nature of the allergens and to assess their clinical relevance