In 1986, Tomaszewski made the following conjecture. Given $n$ real numbers $a_{1},...,a_{n}$ with $\sum_{i=1}^{n}a_{i}^{2}=1$, then of the $2^{n}$ signed sums $\pm a_{1} \pm ... \pm a_{n}$, at least half have absolute value at most $1$. Hendriks and Van Zuijlen (2020) and Boppana (2020) independently proved that a proportion of at least $0.4276$ of these sums has absolute value at most $1$. Using different techniques, we improve this bound to $0.46$.
We prove the following conjecture, due to Tomaszewski (1986): Let $X= \sum_{i=1}^{n} a_{i} x_{i}$, where $\sum_i a_i^2=1$ and each $x_i$ is a uniformly random sign. Then $\Pr[|X|\leq 1] \geq 1/2$. Our main novel tools are local concentration inequalities and an improved Berry-Esseen inequality for Rademacher sums.
Let $v_1$, $v_2$, ..., $v_n$ be real numbers whose squares add up to 1. Consider the $2^n$ signed sums of the form $S = \sum \pm v_i$. Boppana and Holzman (2017) proved that at least 13/32 of these sums satisfy $|S| \le 1$. Here we improve their bound to $0.427685$.
The following assertion was equivalent to a conjecture proposed by B. Tomaszewski : Let $C$ be an $n$-dimensional unit cube and let $H$ be a plank of thickness $1$, both are centered at the origin, then no matter how to turn the cube around, $C\cap H$ contains at least half of the cube's $2^n$ vertices. A lower bound for the number of the vertices of $C$ in $C\cap H$ was obtained.
A recent breakthrough of Conlon and Ferber yielded an exponential improvement on the lower bounds for multicolor diagonal Ramsey numbers. In this note, we modify their construction and obtain improved bounds for more than three colors.
A mixed graph $G$ can contain both (undirected) edges and arcs (directed edges). Here we derive an improved Moore-like bound for the maximum number of vertices of a mixed graph with diameter at least three. Moreover, a complete enumeration of all optimal $(1,1)$-regular mixed graphs with diameter three is presented, so proving that, in general, the proposed bound cannot be improved.
A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we obtain improved error bounds for the number of irreducible polynomials and self-reciprocal irreducible monic polynomials with prescribed coefficients over a finite field. The improved bounds imply that self-reciprocal irreducible monic polynomials with degree $2d$ always exist provided that the number of prescribed leading coefficients is slightly less than $ d/2$.
We prove the first $χ$-bounding function for circle graphs that is optimal up to a constant factor. To be more precise, we prove that every circle graph with clique number at most $ω$ has chromatic number at most $2ω\log_2 (ω) +2ω\log_2(\log_2 (ω)) + 10ω$.
Joret et al. proved that posets with cover graphs of treewidth at most 2 have dimension at most 1276. Their proof is long and very complex. We give a short and much simpler proof that the dimension of such posets is at most 12.
We prove that, if $B_1, \dots, B_n$ are disjoint bases of a rank-$n$ matroid, then there are at least $\lfloor{\frac{n}{6 \lceil{\log n}\rceil}}\rfloor$ disjoint transversals of $(B_1, \dots, B_n)$ that are also bases.
Hadwiger conjectured in 1943 that for every integer $t \ge 1$, every graph with no $K_t$ minor is $(t-1)$-colorable. Kostochka, and independently Thomason, proved every graph with no $K_t$ minor is $O(t(\log t)^{1/2})$-colorable. Recently, Postle improved it to $O(t (\log \log t)^6)$-colorable. In this paper, we show that every graph with no $K_t$ minor is $O(t (\log \log t)^{5})$-colorable.
In this paper an improved bound on the chromatic number of the Pancake graph $P_n, n\geqslant 2$, is presented. The bound is obtained using a subadditivity property of the chromatic number of the Pancake graph. We also investigate an equitable coloring of $P_n$. An equitable $(n-1)$-coloring based on efficient dominating sets is given and optimal equitable $4$-colorings are considered for small $n$. It is conjectured that the chromatic number of $P_n$ coincides with its equitable chromatic number for any $n\geqslant 2$.
The burning number conjecture states that the burning number of a connected graph is at most $\lceil \sqrt{n} \rceil.$ While the conjecture is unresolved, Land and Lu proved that the burning number of a connected graph is at most $ \sqrt{(3/2)n}+O(1).$ Using an algorithmic approach, we provide an improved upper bound for the burning number of a connected graph: $$\bigg \lceil \frac{\sqrt{12n+64}+8}{3} \bigg \rceil = \sqrt{(4/3)n} +O(1) .$$
We show that there exists an absolute constant $c > 0$, such that, for any finite set $A$ of quaternions, \[ \max\{|A+A, |AA| \} \gtrsim |A|^{4/3 + c}. \] This generalizes a sum-product bound for real numbers proved by Konyagin and Shkredov.
Let $S$ be a set of $n\times n$ matrices over a field $\mathbb{F}$. We show that the $\mathbb{F}$-linear span of the words in $S$ of length at most $$2n\log_2n+4n$$ is the full $\mathbb{F}$-algebra generated by $S$. This improves on the $n^2/3+2/3$ bound by Paz (1984) and an $O\left(n^{1.5}\right)$ bound of Pappacena (1997).
Ellenberg and Gijswijt gave the best known asymptotic upper bound for the cardinality of subsets of $\mathbb F_q^n$ without 3-term arithmetic progressions. We improve this bound by a factor $\sqrt{n}$. In the case $q=3$, we also obtain more explicit upper bounds for the Cap Set Problem.
The main purpose of this paper is to prove that the point-line incidence bound due to Vinh (2011) over arbitrary finite fields can be improved in certain ranges by using tools from the VC-dimension theory. As consequences, a number of applications will be discussed in detail.
We generalize upper bounds for constant dimension codes containing a lifted maximum rank distance code first studied by Etzion and Silberstein. The proof allows to construct several improved codes.
We give an improved analysis of the best-of-many Christofides algorithm with lonely edge deletion, which was proposed by Sebő and van Zuylen [2016]. This implies an improved upper bound on the integrality ratio of the standard LP relaxation for the $s$-$t$-path TSP.
We provide a new lower bound on the length of the longest cycle of the binomial random graph $G(n,(1+ε)/n)$ that holds w.h.p. for all $ε=ε(n)$ such that $ε^3n\to \infty$. In the case $ε\leq ε_0$ for some sufficiently small constant $ε_0$, this bound is equal to $1.581ε^2n$ which improves upon the current best lower bound of $4ε^2n/3$ due to Luczak.
In this paper we investigate the integrality ratio of the standard LP relaxation for the metric $s-t$ Path TSP. We make a near-optimal choice for an auxiliary function used in the analysis of Traub and Vygen which leads to an improved upper bound on the integrality ratio of 1.5273.
Recently, Ben Green proved that the two-color van der Waerden number $w(3,k)$ is bounded from below by $k^{b_0(k)}$ where $b_0(k) = c_0\left(\frac{\log k }{\log \log k}\right)^{1/3}$. We prove a new lower bound of $k^{b(k)}$ with $b(k) = \frac{c\log k}{\log \log k}$. This is done by modifying Green's argument, replacing a complicated result about random quadratic forms with an elementary probabilistic result.
Permutation arrays under the Chebyshev metric have been considered for error correction in noisy channels. Let $P(n,d)$ denote the maximum size of any array of permutations on $n$ symbols with pairwise Chebyshev distance $d$. We give new techniques and improved upper and lower bounds on $P(n,d)$, including a precise formula for $P(n,2)$.
We propose and prove an existential theorem for entanglement-assisted asymmetric quantum error correction. Then we demonstrate its superiority over the conventional one.
