An orthodox dominating set on an $n\times n$ chessboard occupies every row of one parity and every column of a possibly different parity. For $n\ge400001$, every such set contains more than $(1/2+1/80000)n-2$ queens, on odd and even boards and with attacking or boundary queens allowed. Second-moment estimates and an exact rational line-weight certificate give a stronger bound for $p$-covers; finite diagonal completion and board extension transfer it to orthodox covers. The cost of extending an arbitrary dominating set to an orthodox cover yields an inequality with an explicit defect term. The previously constructed independent, border-free Type-A $1$-cover of $Q_{221}$ with $111$ queens supplies a finite seed. Classical amplification gives ordinary and independent domination upper bounds with coefficients $112/221$ and $113/221$, respectively. Its order 221 is below the density threshold 400001. For admissible seeds whose orders tend to infinity, the lower limit of these coefficients is at least $1/2+1/16000$.
We construct an infinite family of solutions to the Mondrian art problem with absolute defect bounded above by $280$, of side lengths $n_0 = 315$, $n_1 = 106785$, and $n_{j+2} = 340n_{j+1} - n_j$, where $n_j$ is congruent to $315 \pmod{630}$. This absolute bound shows that the minimum defect does not necessarily increa...
For every positive integer $n$, we prove that there is a constant $C_n$, depending only on $n$, such that $Vol(-K_X) \leq C_n \epsilon^{-(2^n-n-1)}$ whenever $X$ admits an $\epsilon$-lc log Fano boundary. The exponent is optimal in every dimension at least two, even for toric Fano varieties of Picard number one. In par...
The $2\times3$ magic rectangle is a nonlocal game in which two players match entries subject to incompatible parity constraints. We prove that its quantum value is $(1+\sqrt{2/3})/2$ and classify all optimal strategies in arbitrary finite local dimensions, allowing general measurements. Every optimal strategy contains...
Isaac Barouch Essayag, Aryeh Lev Zabokritskiy· 0 citations
For a positive integer $n$, let $S(n)=\{\sigma(d): d\mid n\}$ be ordered by divisibility, and let $a(n)$ be its width. We study how much of the divisor lattice of $n$ survives under the map $d\mapsto \sigma(d)$. For every fixed prime-exponent pattern, the largest possible value of $a(n)$ is the width of a corresponding...
For positive integers $n,r,t$, let $\delta(n,r,t)$ denote the maximum possible minimum degree of a balanced $r$-partite graph with parts of size $n$ and chromatic number at most $t$. Lo, Treglown and Zhao established a general upper bound for this parameter and used it, together with explicit constructions, to determin...
For even $n$, let $P_n$ be the probability that independent uniform strict preference lists on $n$ participants admit a stable perfect matching. We prove \[ P_n\sim\frac{e\,2^{1/4}\Gamma(3/4)}{\sqrt\pi}\,n^{-1/4}. \] This establishes Mertens's conjectured exponent of decay, with a leading constant different from his or...
Caden Young, Ian Studebaker· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
