Related papers: Colorful fractional Helly theorem via weak saturat…
Given a graph $F$ and a positive integer $n$, the weak $F$-saturation number $\mathrm{wsat}(K_n,F)$ is the minimum number of edges in a graph $H$ on $n$ vertices such that the edges missing in $H$ can be added, one at a time, so that every…
We prove that fractional Helly and $(p,q)$-theorems imply $(\aleph_0,q)$-theorems in an entirely abstract setting. We give a plethora of applications, including reproving almost all earlier $(\aleph_0,q)$-theorems about geometric…
The weak saturation number $\mathrm{wsat}(n,F)$ is the minimum number of edges in a graph on $n$ vertices such that all the missing edges can be activated sequentially so that each new edge creates a copy of $F$. A usual approach to prove a…
Van der Waerden's (VDW) colouring theorem in combinatoric number theory [1] has scope for physical applications.The solution of the two colour case has enabled the construction of an explicit mapping of an infinite, one dimensional…
The partial coloring method is one of the most powerful and widely used method in combinatorial discrepancy problems. However, in many cases it leads to sub-optimal bounds as the partial coloring step must be iterated a logarithmic number…
A Helly-type theorem for diameter provides a bound on the diameter of the intersection of a finite family of convex sets in $\mathbb{R}^d$ given some information on the diameter of the intersection of all sufficiently small subfamilies. We…
We consider quantitative versions of Helly-type questions, that is, instead of finding a point in the intersection, we bound the volume of the intersection. Our first main geometric result is a quantitative version of the Fractional Helly…
Given two hypergraphs $G$ and $H$, the weak saturation number $\operatorname{\mathrm{wsat}}(G,H)$ is the minimum number of edges in a spanning subhypergraph $F$ of $G$ such that the missing edges of $F$ can be added one at a time so that…
We prove a colorful extension of a Helly-type theorem by Danzer and Gr\"{u}nbaum (Combinatorica, 1982) concerning two-piercing families of axis-parallel boxes in $\mathbb{R}^d$. We also show that our result is tight by constructing extremal…
We prove general topological Radon-type theorems for sets in $\mathbb R^d$ or on a surface. Combined with a recent result of Holmsen and Lee, we also obtain fractional Helly theorem, and consequently the existence of weak $\varepsilon$-nets…
Weak coloring numbers generalize the notion of degeneracy of a graph. They were introduced by Kierstead \& Yang in the context of games on graphs. Recently, several connections have been uncovered between weak coloring numbers and various…
One widely applied sufficient condition for the existence of a colorful simplex in a vertex-colored simplicial complex is a topological extension of Hall's transversal theorem due to Aharoni, Haxell, and Meshulam. We prove a similar…
The classical Halpern-L\"auchli theorem states that for any finite coloring of a finite product of finitely branching perfect trees of height $\omega$, there exist strong subtrees sharing the same level set such that tuples consisting of…
Aharoni, Berger and Ziv recently proved the fractional relaxation of the strong colouring conjecture. In this note we generalize their result as follows. Let $k\geq 1$ and partition the vertices of a graph $G$ into sets $V_1,..., V_r$, such…
Various results ensure the existence of large complete bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind,…
This paper is concerned with global existence of weak solutions for a weakly dissipative $\mu$HS equation by using smooth approximate to initial data and Helly$^{,}$s theorem.
The local properties problem of Erd\H{o}s and Shelah generalizes many Ramsey problems and some distinct distances problems. In this work, we derive a variety of new bounds for the local properties problem and its variants. We do this by…
Let $K$ be the scalar field of real numbers or complex numbers and $L^{0}(\mathcal{F},K)$ the algebra of equivalence classes of $K-$valued random variables defined on a probability space $(\Omega,\mathcal{F},P)$. In this paper, we first…
L. Soukup formulated an abstract framework in his introductory paper for proving theorems about uncountable graphs by subdividing them by an increasing continuous chain of elementary submodels. The applicability of this method relies on the…
We prove the following variant of Helly's classical theorem for Hamming balls with a bounded radius. For $n>t$ and any (finite or infinite) set $X$, if in a family of Hamming balls of radius $t$ in $X^n$, every subfamily of at most…