This note propose a counterexample to the Anstee--Sali conjecture for forbidden configurations. The basic candidate is the $4$-uniform family \[ F_2=\{xyab,xybc,xycd,xyda\} \] on six vertices: a fixed two-vertex core $\{x,y\}$ joined to the four edges of a $4$-cycle. We give an explicit certificate that every four-fold product whose factors are of type $I$, $I^c$, or $T$ contains $F_2$, while $I^3$ avoids it. Thus, $X(F_2)=4$, so the conjecture predicts $\operatorname{forb}(m,F_2)=\Theta(m^3)$. On the other hand, a result of Mubayi on complete multipartite hypergraphs implies \[ \operatorname{forb}(m,F_2)=\Omega(m^{7/2}), \] which is asymptotically larger than $m^3$. The example was found by GPT-5.6 Sol.
Let $G= C_{n_1}\oplus\cdots\oplus C_{n_r}$ be a finite abelian group with $1<n_1\mid\cdots\mid n_r$, and let $\rr(G)=r$ denote its rank. The Davenport constant $\DD(G)$ is the least integer $\ell$ such that every sequence of $\ell$ elements of $G$ contains a nonempty zero-sum subsequence, and $\DD^*(G)=1+\sum_{i=1}^r(n...
A longstanding fundamental problem of Burr, Erd\H{o}s, Faudree, Rousseau and Schelp (\emph{Trans. Amer. Math. Soc.}, 1982) is to determine the exact value of the least integer $f(m)$, for odd $m\ge3$, such that every tree $T_n$ on $n\ge f(m)$ vertices satisfies $R(T_n,C_m)=2n-1$. We settle this problem for all sufficie...
Given graphs $H$ and $F$, the generalized Tur\'{a}n number ${\rm ex}(n,H,F)$ is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Alon and Shikhelman (J. Combin. Theory Ser. B, 2016) initiated the systematic study of generalized Tur\'{a}n problems. Recently, Gao, Wu and Xue (J. Graph Theory, 2026) as...
Albertson's conjecture asserts that every finite simple graph $G$ with $\chi(G) \ge r$ satisfies $\operatorname{cr}(G) \ge \operatorname{cr}(K_r)$. Building on Cranston's verification for $r \le 24$ and his reduction of $r \in \{25,26\}$ to three residual orders, we eliminate those residual cases and then prove the cas...
We disprove Kusner's 1983 conjecture that every equilateral set in $\ell_p^n$ with $2<p<\infty$ has at most $n+1$ points: there exist $58$ points in $\mathbb{R}^{56}$ whose pairwise $\ell_5$ distances are all equal, so the maximum equilateral-set size satisfies $e(\ell_5^{56})\ge58>57$. This is the first equilateral se...
Let $\mathbb{P}(x\prec y)$ be the probability that $x$ precedes $y$ in a uniformly random linear extension of an $n$-element poset $P$, and define the balancing coefficient to be $\delta(x,y)=\min(\mathbb{P}(x\prec y),\mathbb{P}(y\prec x))$ with $\delta(P)=\max_{x,y}\delta(x,y)$. We prove (Theorem 1) that sufficiently...
Max Aires· 1 citation· ⚡1
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
