arXiv subjects
Publications and source records attributed to Skand Parvatikar.
We show that for every $n$ unit vectors $v_1, \dots, v_n$ in the $d$-dimensional Euclidean space, there exist signs $\varepsilon_1, \dots, \varepsilon_n \in \{\pm 1\}$ such that $\lVert \varepsilon_1 v_1 + \dots + \varepsilon_n v_n \rVert \ge \sqrt{2n - d}$, and we characterize the equality cases.
In this paper, we obtain new upper and lower bounds on the colorful Helly number and the Tverberg number in an abstract convexity space with Radon number $r$. We prove an upper bound of $(r-1)2^r$ on the colorful Helly number, which is a factor $O(r)$ far from the lower bound, $2^{r-1}-1$. The best previous bound, by Holmsen and Lee (2021), was $r^{r^{\log r}}$. As a consequence, we obtain improved quantitative bounds for fractional Helly numbers, the selection lemma, weak $\varepsilon$-nets, and the $(p,q)$-theorem, in abstract convexity spaces. Furthermore, using the improved colorful Helly bound, we prove that the Tverberg number $r_k$ is at most $O(r^{\lceil \log_2 r \rceil})k$. The best previous bound, by Pálvölgyi (2022), was $r^{r^{r^{\log r}}}k$. We also study the $t$-wise Tverberg number $r_{k,t}$, which is the least $\ell$ for which any $\ell$ points can be divided into $k$ parts such that the convex hulls of any $t$ parts intersect. We prove the optimal bound $r_{k,t} = O_t(kr)$, in any $S_4$ separable space. In the other direction, we construct a separable space in which $r_k = Θ(r^2 k)$, while $r_{k,2}=Θ(rk)$. This proves that the weak version of Eckhoff's conjecture, which suggested that $r_k=O(rk)$ in any abstract convexity space, fails even in separable spaces. In addition, this shows that the abstract analogue of Reay's conjecture (1979), suggesting that $r_{k,2}=r_k$ in Euclidean spaces, already fails in separable convexity spaces.
Let $G = (V,E)$ be an edge-colored graph, and let $δ^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$. We show that for a sufficiently large $n$ if $G$ is an edge-colored triangle-free graph of order $n$ that satisfies $δ^c(G)\geq (n+7)/5$, then $G$ contains a rainbow cycle of length four, which improves a bound of Ding et al. and is best possible. In addition, we show that given $k$, there is $n_0$ such that for $n\geq n_0$, if $G$ is an edge-colored triangle-free graph with $δ^c(G)> n/5+3$, then $G$ contains a rainbow cycle of length $4k$.