Mathematics and Statistics

A question of Erdős asked whether there exists a set of 𝑛 points such that 𝑐 ⋅ 𝑛 distances occur more than 𝑛 times. We provide an affirmative answer to this question, showing that there exists a set of 𝑛 points such that $⌊\frac{n}{4}⌋$ distances occur more than 𝑛 times. We also present a generalized version, finding a set of 𝑛 points where 𝑐_𝑚 ⋅ 𝑛 distances occurring more than 𝑛 + 𝑚 times.

The Erdős Matching Conjecture states that the maximum size 𝑓 (𝑛, 𝑘, 𝑠) of a family $F\subseteq \left(\begin{array}{l}\left[n\right]\\ k\end{array}\right)$ that does not contain 𝑠 pairwise disjoint sets is max. $\left\{\left|A_{k,s}\right|,\left|B_{n,k,s}\right|\right\}$ , where $A_{k,s}=\left(\left[\begin{array}{c}sk-1\\ k\end{array}\right]\right)$ and $B_{n,k,s}=\left\{B\in \left(\begin{array}{c}\left[n\right]\\ k\end{array}\right):B\cap \left[s-1\right]\ne \varnothing \right\}$ . The case 𝑠 = 2 is simply the Erdős-Ko-Rado theorem on intersecting families and is well understood. The case 𝑛 = 𝑠𝑘 was settled by Kleitman and the uniqueness of the extremal construction was obtained by Frankl. Most results in this area show that if 𝑘, 𝑠 are fixed and 𝑛 is large enough, then the conjecture holds true. Exceptions are due to Frankl who proved the conjecture and considered variants for 𝑛 ∈ [𝑠𝑘, 𝑠𝑘 + 𝑐𝑠,𝑘 ] if 𝑠 is large enough compared to 𝑘. A recent manuscript by Guo and Lu considers non-trivial families with matching number at most 𝑠 in a similar range of parameters.

In this short note, we are concerned with the case 𝑠 ≥ 3 fixed, 𝑘 tending to infinity and 𝑛 ∈ {𝑠𝑘, 𝑠𝑘 + 1}. For 𝑛 = 𝑠𝑘, we show the stability of the unique extremal construction of size $\left(\begin{array}{c}sk-1\\ k\end{array}\right)=\frac{s-1}{s}\left(\begin{array}{c}sk\\ k\end{array}\right)$ with respect to minimal degree. As a consequence we derive $\underset{k\to \infty }{\lim }\frac{f\left(sk+1,k,s\right)}{\left(\begin{array}{c}sk+1\\ k\end{array}\right)}<\frac{s-1}{s}-{\epsilon }_s$ for some positive constant 𝜀_𝑠 which depends only on 𝑠.

A long standing Total Coloring Conjecture (TCC) asserts that every graph is total colorable using its maximum degree plus two colors. A graph is type-1 (or type-2) if it has a total coloring using maximum degree plus one (or maximum degree plus two) colors. For a graph 𝐺 with 𝑚 vertices and for a family of graphs {𝐻₁, 𝐻₂, … , 𝐻_𝑚}, denote $G\tilde{\circ }{\Lambda }_{i=1}^m{H}_i$ , the generalized corona product of 𝐺 and 𝐻₁, 𝐻₂, … , 𝐻_𝑚. In this paper, we prove that the total chromatic number of $G\tilde{\circ }{\Lambda }_{i=1}^m{H}_i$ is the maximum of total chromatic number of 𝐺 and maximum degree of $G\tilde{\circ }{\Lambda }_{i=1}^m{H}_i\in$ plus one. As an immediate consequence, we prove that $G\tilde{\circ }{\Lambda }_{i=1}^m{H}_i$ is type-1 when 𝐺 satisfies TCC and also the corona product of 𝐺 and 𝐻 is type-1 if 𝐺 satisfies TCC. This generalizes some results in (R. Vignesh. et. al. in Discrete Mathematics, Algorithms and Applications, 11(1): 2019) and all the results in (Mohan et. al. in Australian Journal of Combinatorics, 68(1): 15-22, 2017.)

We study the property of Kelley and the property of Kelley weakly on Hausdorff continua. We extend results known for metric continua to the class of Hausdorff continua. We also present new results about these properties.

The aim of this paper is to study the interrelationship between various forms of (F, G)-shadowing property and represent it through the diagram. We show that asymptotic shadowing is equivalent to (ℕ₀, F _𝑐𝑓 )-shadowing property and that (ℕ₀, F _𝑐𝑓 )-shadowing implies (F _𝑐𝑓 , F _𝑐𝑓 )-shadowing. Necessary examples are discussed to support the diagram. We also give characterization for maps to have the (F, G)-shadowing property through the shift map on the inverse limit space. Further, we relate the (F, G)-shadowing property to the positively F _𝑠-expansive map. Also, we obtain the necessary and sufficient condition for the identity map to have (ℕ₀, F _𝑡)-shadowing property.

We prove that the filtered GRID invariants of Legendrian links in link Floer homology, and consequently their associated invariants in the spectral sequence, obstruct decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on ℝ³, strengthening a result by Baldwin, Lidman, and the fifth author.

In [3] it is shown, answering a question of Jordán and Nguyen [9], that universal rigidity of a generic bar-joint framework in ℝ¹ depends on more than the ordering of the vertices. The graph 𝐺 that was used in that paper is a ladder with three rungs. Here we provide a general answer when that ladder with three rungs in the line is universally rigid and when it is not.

In this paper the author studies the problem of finding the farthest points in an intersection of balls to a given point 𝐶₀. A polynomial algorithm is presented which solves the problem under the conditions that the given point is outside of the convex hull of the balls centers. It is shown that in this particular case the problem of finding the smallest ball centered in 𝐶₀ which includes the intersection of balls is actually convex.

In this article, we present new results on specific cases of a general Young integral inequality established by Páles in 1990. Our initial focus is on a bivariate function, defined as the product of two univariate and separable functions. Based on this, some new results are established, including particular Young integral-type inequalities and some upper bounds on the corresponding absolute errors. The precise role of the functions involved in this context is investigated. Several applications are presented, including one in the field of probability theory. We also introduce and study reverse variants of our inequalities. Another important contribution is to link the setting of the general Young integral inequality established by Páles to a probabilistic framework called copula theory. We show that this theory provides a wide range of functions, often dependent on adjustable parameters, that can be effectively applied to this inequality. Some illustrative graphics are provided. Overall, this article broadens the scope of bivariate inequalities and can serve related purposes in analysis, probability and statistics, among others.

Let 𝑛, 𝑠, 𝑣 be positive integers and F ⊂ 2^[𝑛]. Suppose that the union of any 𝑠 sets of F has size at most 𝑠𝑣 and 𝑛 ≥ 2^𝑠+3𝑣. The main result implies the best possible bound $\left|F\right|\le \left(\begin{array}{c}n\\ v\end{array}\right)+\left(\begin{array}{c}n\\ v-1\end{array}\right)+\dots +\left(\begin{array}{c}n\\ 0\end{array}\right)$ . For 𝑛 ≤ (2^𝑠 − 𝑠 − 1)𝑣 the same statement is no longer true. Several statements of a similar flavor are established as well, providing further evidence for an old conjecture of the first author.