Let \(H\) be a finite hypergraph with rank $r$ and minimum degree \(d\). In the Maker--Breaker degree game, Maker and Breaker alternately claim previously unclaimed hyperedges of \(H\), with Maker moving first, and Breaker seeks to maximize the minimum degree $\delta(H_{\mathrm B})$ of his spanning subhypergraph $H_{\m...
In this short note, we point out that the recent lower bounds for off-diagonal Ramsey numbers $r(s,t)$ with fixed $s\ge 5$, together with the recent proof of the Ajtai--Erd\H{o}s--Koml\'os--Szemer\'edi independence conjecture, imply \[ r(4,t)=\frac{t^3}{(\log t)^{2+o(1)}}. \] This note is solely for informal circulatio...
For $n\ge1$, let $F(n)$ be the least $H$ such that any $H$ consecutive integers contain $n$ pairwise distinct integers $a_1, a_2, \dots, a_n$ with $k \mid a_k$ for $1\le k\le n$, and define $h_{\mathbb P}(n)$ analogously for the primes at most $n$. We prove \[ F(n)\le n^{4/3}\exp\!\left(O\!\left(\frac{\log n}{\log\log...
Kaizhe Chen, Samuel Korsky· 0 citations
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