A weighing design W(m, z)k in the sense of Lejeune Herman and Goos (arXiv:2608.04814) is an m × z matrix over {0, ± 1} with exactly k non-zero entries in every column, at most m-k zeros in every row, and W^(T)W=kI_z. Their Table 1 counts the isomorphism cla…
For a finite set A of integers let hA be the set of sums of h not necessarily distinct elements of A, and let R(h, k)={|hA|: A ⊆ ℤ, |A|=k}. Nathanson determined R(h, 3) for every h and wrote that for k=4 "the problem is still open: Compute R(h, 4) ⊆ [3h+1,C…
Campbell, Currie and Rampersad recently introduced the reduced factor and abelian complexity functions of an infinite word: length-n factors are counted after every maximal block of equal letters has been collapsed to a single letter, up to equality (respec…
Karve and Hirani attach to a hypergraph g the set of CNFs that live on it — one clause per edge, on exactly that edge's variables, with an arbitrary choice of signs — and call g totally satisfiable when every such CNF is satisfiable. For the complete a-unif…
In arXiv:2607.21761v1, Conant and Terry introduced a notion of non-uniform ladder for a function f: X × Y → [0,1] and proved a real-valued analogue of the Shelah–Hodges tree/ladder correspondence: if f admits a (C(2k, k)-1, 2δ)-tree, then f admits a (k,δ)-l…
For a>0 and s ∈ {0,…,n-1} let the generalized corner vector v_(a,s) ∈ 𝕊ⁿ⁻¹ be the unit vector along (a,1,…,1,0,…,0) with s ones. Its hyperoctahedral orbit v_(a,s)^(Bₙ) is called proper when a ≠ 1; a weighted spherical t-design which is a union of such orbi…
Baker, Huh, Kummer and Lorscheid attach to a matroid M a threshold q(M) and conjecture exact values for q(n)=q(U_(2,n)). Ercan has recently settled the first two open cases, q(6)=log₂9/4 and q(7)=1, and his proof has exactly one computer-assisted step: four…
For a graph G of order n, the upper oriented general position number ugpop(G) is the largest general position number of an orientation of G. Chandran S. V., Di Stefano, Erskine, Haritha S., Thomas and Tuite proved that ugpop(G) equals the largest order of a…
For a graph G of order n with adjacency matrix A let F_G(t) be the average of |(e^(-i tA))ᵥᵤ|² over ordered pairs of distinct vertices. Song recently determined the maximum of the dense-scaling limit of n²F_G(τ/n) over triangle-free graphons: the balanced c…
A quantum Latin square of order n is an n × n array of unit vectors of ℂⁿ every row and every column of which is an orthonormal basis; its cardinality is the number of entries once vectors differing by a global phase are identified. Zhang, Lv and Cao tabula…
Two graphs are quantum isomorphic when the isomorphism game between them has a perfect quantum-commuting strategy. Such a pair need not be isomorphic: Atserias, Mančinska, Roberson, Šámal, Severini and Varvitsiotis turned the Mermin magic square into a quan…
A quantum Latin square of order n is an n × n array of unit vectors of ℂⁿ each of whose rows and columns is an orthonormal basis; its cardinality is the number of entries up to a global phase, and it has maximum cardinality when that number is n². Huang and…
Tokushige, in his work on Alon's transmitting problem and a multicolour Beck–Spencer lemma, proves ⌊(1-1/q)n⌋+1 ≤ b(H(n, q)) ≤ ⌊(1-1/q)n+(q+1)/2⌋ for the burning number of the Hamming graph with q ≥ 3, conjectures that the upper bound is the truth, and clos…
Let f(n) be the number of ways to partition the vertex set of the n-cube into subcubes — OEIS A018926, the function f(d) of Alon, Balogh and Potapov, and the (n+1)-st cubical Bell number B^((c))ₙ₊₁ of Duarte and Solus, which counts the stagings of a level o…
The Zarankiewicz number z(m,n;s,t) is the largest number of ones in an m × n zero–one matrix with no all-ones s × t submatrix. For s=t=3 the exact values are known only in a small range, and the state of the art is a table of lower and upper bounds that sev…
An antichain C in a finite lattice L is Boolean if bigwedge S vee bigwedge S' = bigwedge (S ∩ S') for all S,S' ⊆ C; these antichains index the perfect modules over the incidence algebra of L. Garber, Goltermann, Horiatakis, König and Gottesman count the Boo…
Put n=2ᵏ-1 points evenly on a circle. A permuted copy is a k-subset whose cyclic gaps are a permutation of 1,2,4,…,2ᵏ⁻¹; Stromquist conjectured that every two-colouring of the n points contains a monochromatic permuted copy, and Damásdi, Frankl, Pach and Pá…
A signed difference set in a finite abelian group G of order v is an element D=Σ_(g)a_g g of ℤ[G] with a_g ∈ {0, ± 1}, support size k, satisfying DD^((-1))=(k-λ) 0_G+λ G. He and Wu recently produced a (125,28,3) signed difference set in C₅³ out of quartic m…
Isham, Lakhotia, Monroe and Petrini recently proved that the polarity quotient of a Kronecker product of generalized polygons is the Kronecker product of their polarity quotients, and deduced the existence of diameter-three graphs of degree 18, 19 and 20 on…
Wilhelm has refuted the Non-Cancelling-Intersections conjecture of Amarilli, Monet and Suciu by showing that for every prime p ≥ 10⁵ some marking m of the lines of 𝔽ₚ² makes the lattice P_(p,m), of p³+2p²+2 elements, admit no winning dot-algebra tree; the…
For a matroid M and a building set G of its lattice of flats, the Chow polynomial H(M,G)(t) is palindromic, and Coron, Ferroni and Li proved that its γ-vector is nonnegative when (M,G) is flag, and is the f-vector of a simplicial complex when (M,G) is compl…
For a simple graph G let L²(G) be its second iterated line graph, and for integers d ≥ 3 and k ≥ 1 let G(d,k) be the class of simple graphs with δ(G)=d and κ'(G)=k, so that κ_(L²)(d,k)=inf{κ(L²(G)):G ∈ G(d,k)}. A recent preprint of Zou, Xiong, Zhan and Lai…
For a partition λ of d let h_λ be the product of complete homogeneous symmetric polynomials in n variables and H_(n,λ)=h_λ/h_λ(1ⁿ) its term-normalised form. Cuttler, Greene and Skandera proved that dominance of partitions implies H_(n,λ) ≤ H_(n,μ) on the no…
Soberón has settled the last open case of Grünbaum's 1960 equipartition problem by producing a mass in ℝ⁴ that no four hyperplanes equipartition. The computational core of that proof is a system of five polynomials f₀,…,f₄ in six variables, of degree at mos…
