Let $h$ be a positive integer, and let $A$ be a finite set of integers. We derive an exact formula for $|hA|$. Furthermore, let $A=\{0,1,\ldots,s,a,b\}$, $1\leq s<a<b$, and write $b=qa+r$ with $0\leq r<a$. By using generating function, we prove that $|hA|$ equals a definite explicit formula expressed in terms of certain truncated binomial coefficients for all positive integers $h$ if and only if $r=0$ or $qs+r\geq a$. This generalizes a result of Nathanson.
Let $\mathbb{N}_0=\{0,1,2,\ldots\}$ and let $h\ge2$ be an integer. For a set $A\subseteq\mathbb{N}_0$, write $hA$ for the set of all sums of $h$, not necessarily distinct, elements of $A$. In this paper, we prove that for every $h\ge2$, there is a partition $\mathbb{N}_0=A\sqcup B$ such that $A$ is a minimal asymptotic basis of order $h$ and $B$ is a maximal asymptotic nonbasis of order $h$. This solves an open problem posed by Nathanson in 1974.
Let $K$ be a number field of degree $D$ with maximal order $\mathcal{O}_K$. We show that under certain conditions on $K$, which in particular are always satisfied if $D$ is odd or if $D \geq 3$ and $K$ is primitive, the set of positive integers $N_K$ that can be expressed as a sum of two units in $\mathcal{O}_K^*$ is a finite effectively computable set. This result partially resolves an open problem posed by Tinkov\'{a}, Yatsyna, and the first author. We illustrate our method by explicitly computing $N_K$ for the smallest totally real quintic field $K$ with Galois group $S_5$.
Let $X$ be an alphabet, let $t\geq 0$ and $n\geq t+1$, and let $((a_i,b_i))_{i=1}^{m}$ be an ordered family of word pairs in $X^n$ satisfying $dist(a_i,b_i)\geq t+1$ for every $i$ and $dist(a_i,b_j)\leq t$ whenever $i<j$. We prove the sharp bound $m\leq 2^{t+1}$, thereby resolving a problem posed by Alon, Jin, and Sudakov. Our proof uses a linear-algebraic method based on a characteristic-two algebra, which may be of independent interest.
Let $P$ be a finite nonempty poset with $n$ elements, let $f:P\to\{1,\ldots,n\}$ be a uniformly random order-preserving bijection, and put $h_P(x)=\mathbb{E}[f(x)]$. Aires and Kahn (2025) introduced $\operatorname{gap}(P)$ as the largest difference between consecutive values in the ordered list consisting of $0$, $n+1$, and all the expected ranks $h_P(x)$. Write ${w}(P)$ for the largest size of a pairwise incomparable subset. We prove three results. First, we prove a weighted strengthening of an ideal inequality conjectured by Kahn and obtain the explicit gap-width bound $\operatorname{gap}(P)\le 2 {w}(P)-1$. Second, for every $L>0$ we construct a width-two poset such that the expected-rank list of every maximal chain has a gap of at least $L$, with $0$ and $|P|+1$ added as endpoints. Finally, for every $r\in\mathbb{N}$, we construct a poset $P_r$ for which the relative order induced on every nonempty selected set $X$ has base-two entropy below $3|X|$, while $\operatorname{gap}(P_r)\ge(3/2)^r$. Thus the gap can be arbitrarily large while the induced order on every selected set has relatively small entropy. The key ideas behind all three results were found by ChatGPT 5.6 Sol.
For a finite set $A\subset\mathbb{Z}$, write $hA$ for its $h$-fold sumset, and let \[ R(h,k)=\{|hA|:A\subset\mathbb{Z},\ |A|=k\}. \] We determine the part of $R(h,4)$ lying between $4h+2$ and $6h-4$: for $h=4$ the only value is $5h-1$, while for $h\geq 5$ the only values are $5h-1$ and $5h+1$. This proves Rajagopal's conjectured gap $5h\notin R(h,4)$ for every $h\geq 4$. For $h\geq 6$, it also yields the new missing interval $[5h+2,6h-4]$, which lies outside Rajagopal's general excluded set. Lev's lower bound for the successive growth of multiple sumsets reduces the problem to normalized sets of affine diameter five, of which there are only six. Reflection and four elementary exact sumset computations finish the classification.
For a graph $H$ with $3\mid e(H)$, the zero-sum Ramsey number $R(H,\Z_3)$ is the least integer $N$ such that every labeling of the edges of $K_N$ by elements of $\Z_3$ contains a copy of $H$ whose edge labels sum to zero. We determine the last previously unresolved infinite family in the complete-graph case modulo $3$. More precisely, we prove that \(R(K_n,\Z_3)=n+3\) for every $n\ge 10$ satisfying $n\equiv 1\pmod 3$. Consequently, for $k\ge 1$, \(R(K_{9k+7},\Z_3)=9k+10\), resolving a problem of Caro and Mifsud.
Cheng Chi, Jia-Lin He, Fuhong Ma· 0 citations
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