Fix one colour class in the checkerboard colouring of an $n\times n$ integer grid, and let $M_4(n,\varepsilon)$ be the largest subset having at most two points in every row, column, and diagonal of slopes $\pm1$. We prove the near-saturation bound $M_4(n,\varepsilon)\leq2n-4$ for $n\geq6$. The proof uses first and second moments of the four line families: a hypothetical set of size $2n-3$ produces row, column, and diagonal deficits whose exact moment identities contradict Cauchy--Schwarz. A finite argument handles $n=6$. The same identity extends to arbitrary deficit multisets. It gives $M_4(n,\varepsilon)\leq2n-d$ whenever $d\geq4$ is an integer and $n\geq3d-4$, and an entirely discrete asymptotic estimate \[ M_4(n,\varepsilon)\leq(\sqrt{21}-3)n+8. \] We also prove a general continuum-to-discrete theorem for the associated four-direction fractional packing problem. Applying it to the exact continuum dual certificate constructed in earlier work yields, for both colours, \[ L_{\mathrm{mono}}(n,\varepsilon)\leq\alpha n+O(1), \qquad \alpha\approx1.5768233968738, \] and hence the same upper bound for $M_4$ and for checkerboard no-three-in-line sets.
The queen's graph $Q_n$ has the squares of the $n\times n$ chessboard as vertices, with adjacency defined by a common row, column, or diagonal. We give an explicit set of $82$ queens on $Q_{163}$. In centered coordinates, all queen coordinates are odd, every odd row and odd column is occupied exactly once, and the occu...
Let $E_2$ be the extraspecial group of order $3^5$ and exponent three. We prove that $d(E_2\times C_3^r)=2r+10$ for $0\le r\le3$. The upper bounds follow from signed zero-block identities and two finite statements in the four-dimensional symplectic space over $\mathbb F_3$. The first supplies edge weights for all compl...
We study the largest size $C(n)$ of a subset of the $n\times n$ integer grid containing no isosceles triple, including equally spaced collinear triples. We prove $C(n)=\Omega\left(n\sqrt{\log\log n/\log n}\right)$ by applying the sparse hypergraph coloring theorem of Cooper and Mubayi. Elementary geometry and primitive...
Suppose a set of integers $A\subseteq\{1,\dots,N\}$ has no solutions to $a-a'=n\lfloor\sqrt[3]2n\rfloor,$ for distinct $a,a'\in A,$ and $n\in \mathbb{N}.$ We show that $|A|\ll N^{1-c}$ for some absolute constant $c>0.$ To do this, we prove quantitative bounds on the van der Corput property for certain sets of bracket q...
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...
We record a finite-field census of the split-cubic elliptic curves $E_{a,b,c}^{(n)}:y^2=(ax+n)(bx+n)(cx+n)$ attached to positive integral $D(n)$-triples. For $n\in\{-3,3,5,8,12,20\}$ there are $164$ such triples with $1\le a<b<c\le100$, allowing zero squares. Their reductions at the ten primes $\operatorname{NextPrime}...
Sompong Chuysurichay, S. Jaidee, Chatchawan Panraksa et al.· 0 citations
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