A 4AP-free permutation of the positive integers
We construct a permutation of the positive integers containing no four-term arithmetic progression as a subsequence.
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arXiv subjects
Boon Suan Ho
Publications and source records attributed to Boon Suan Ho.
Halving the original Kalton--Roberts upper bound for nearly additive set functions
Let $K_{KR}$ be the least constant governing uniform approximation of approximately additive real-valued set functions by finitely additive signed measures. We prove that $$K_{KR}\le\frac{374167}{20000}=18.70835.$$ Starting from a norming functional for best uniform approximation, we form two families of almost extremal sets with complementary point frequencies. Intersections and biregular expanders then give two recombination schemes. The sharper estimate comes from using the schemes on different ranges and from cancelling the positive and negative errors with unequal weights. This improves the previously published bound $38.8$.
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Counterexamples for lacunary dilates via dyadic spike blocks
We construct dyadic lacunary counterexamples for two problems of Erdős on pointwise behavior of dilates on the circle. The main device is a dyadic spike block: rare positive spikes create long positive runs in the lacunary averages, while a deterministic lower floor prevents cancellation from the remaining stages. The endpoint construction gives a mean-zero $f\in\bigcap_{1\le q<\infty}L^q(\mathbb T)$ and a sequence $n_j=2^{m_j}$, $n_{j+1}/n_j\ge2$, such that $$ \|f-S_Nf\|_2\ll (\log\log N)^{-1/2}, \qquad \limsup_{N\to\infty} \frac1N\sum_{j\le N}f(n_jx)=+\infty $$ for almost every $x$. Thus Matsuyama's positive theorem at exponent $c>1/2$ cannot be extended to the endpoint $c=1/2$, and Erdős Problem #996 has a negative answer. A second choice of parameters gives, for every $2\le p<\infty$, functions $f\in L^p(\mathbb T)$ with $$ \limsup_{N\to\infty} \frac{\sum_{j\le N}f(n_jx)} {N(\log N)^{1/p-\varepsilon}} =+\infty \qquad(\varepsilon>0) $$ almost everywhere; the case $p=2$ answers Erdős Problem #995. We also include a bounded small-set companion construction.
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Erdős's diameter conjecture for separated distances fails in high dimensions
Erdős asked whether every $n$-point set in Euclidean space whose $\binom{n}{2}$ pairwise distances are mutually at least $1$ apart must have diameter at least $(1+o(1))n^2$. We disprove this statement by constructing for every prime power $q$ a set $\mathcal X_q\subset \mathbb R^{q^2+q}$ of $n=q+1$ points such that all pairwise distances in $\mathcal X_q$ are mutually at least $1$ apart, while $$\operatorname{diam}(\mathcal X_q)\le\Bigl(1-\frac{1}{π^2}+o(1)\Bigr)n^2.$$ The proof is fully formalized in Lean 4.
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A new lower bound for the kissing number in 19 dimensions
We prove that the kissing number in 19 dimensions is at least 11948, improving the bound of Cohn and Li by 256. By the odd-sign construction of Cohn and Li, it is enough to find a binary code of length 19 and minimum distance 5 inside the ambient 5-punctured extended binary Golay code. We construct such a code explicitly, of size 1280. The construction is organized around a chain of linear codes $M\le K\le D$, $|M|=64$, $|K/M|=16$, and $|D/K|=4$. The 21 words of $D$ of weight 3 or 4 lie in exactly five nonzero $M$-cosets inside $K$. Those five cosets define a Cayley graph on $K/M\cong\mathbb F_2^4$ with connection set $\{e_1,e_2,e_3,e_4,e_1+e_2+e_3+e_4\}$, hence the Clebsch graph. A 5-coclique in that quotient lifts first to a 320-word code in $K$ and then, by taking all four cosets of $K$ in $D$, to the desired 1280-word code.
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The Ferrers bound for spanning trees in bipartite graphs
We prove Ehrenborg's conjecture that every connected bipartite graph $G$ with parts of size $m$ and $n$ has at most $\frac{1}{mn}\prod_{v\in V(G)} \operatorname{deg}(v)$ spanning trees, and that equality holds if and only if $G$ is a Ferrers graph. The proof is fully formalized in Lean 4.
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3AP-free permutations have no exponential growth rate
Let $θ(n)$ be the number of permutations of $\{1,\dots,n\}$ with no $3$-term arithmetic progressions. We prove that $\lim_{n\to\infty}θ(n)^{1/n}$ does not exist.
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A generalization of Boppana's entropy inequality
In recent progress on the union-closed sets conjecture, a key lemma has been Boppana's entropy inequality: $h(x^2)\geϕxh(x)$, where $ϕ=(1+\sqrt5)/2$ and $h(x)=-x\log x-(1-x)\log(1-x)$. In this note, we prove that the generalized inequality $α_kh(x^k)\ge x^{k-1}h(x)$, first conjectured by Yuster, holds for real $k>1$, where $α_k$ is the unique positive solution to $x(1+x)^{k-1}=1$. This implies an analogue of the union-closed sets conjecture for approximate $k$-union closed set systems. We also formalize our proof in Lean 4.
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Two product formulas for counting successive vertex orderings
A vertex ordering of a graph $G$ is a bijection $π\colon\{1,\dots,|V(G)|\}\to V(G)$. It is successive if the induced subgraph $G[v_{π(1)},\dots,v_{π(k)}]$ is connected for each $k$. Lixing Fang, Hao Huang, János Pach, Gábor Tardos, and Junchi Zuo [J. Comb. Theory A199 (2023), 105776] gave formulas for counting the number of successive vertex orderings for a class of graphs they called "fully regular," and conjectured that these formulas could be written as certain products involving differences or ratios of binomial coefficients in two cases: When the graph is the line graph $L(K_n^{(3)})$ of the complete $3$-uniform hypergraph, or when it is the line graph $L(K_{m,n}^{(1,2)})$ of a complete "bipartite" $3$-uniform hypergraph. In this paper, we confirm both of these conjectures.
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Equality cases for a bound on the chromatic number
It is known that the inequality $$ \frac{χ(G)(χ(G)-1)}{2} + |V| - χ(G) \leq |E|$$ holds for all connected graphs, where $χ(G)$ denotes the chromatic number of $G$. We prove that equality holds whenever the graph consists of a complete graph or an odd cycle, together with finitely many trees attached to its vertices.
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