论文标题
单一方程式的不对称随机RADO定理：$ 0 $陈述
An asymmetric random Rado theorem for single equations: the $0$-statement
论文作者
论文摘要
RADO的著名结果是那些整数矩阵$ a $ a $的特征，即划分的常规分区，即，对正整数的任何有限着色都会产生对等式$ ax = 0 $的单色解决方案。 Aigner-Horev和人最近在二项式随机套件的概率阈值$ [n] _p $具有不对称的随机rado属性：定期矩阵$ a_1，\ dots，a_r $（对于固定的$ r \ f \ egeq 2 $），但是一个$ r $ -colours $ a $ a $ a $ [n]有一个$ i $颜色的解决方案，$ a_i x = 0 $。这概括了由Rödl和Ruciński和Friedgut，Rödl和Schacht解决的对称情况。 Aigner-Horev和人证明了其不对称猜想的$ 1 $陈述。在本文中，我们可以在$ a_i x = 0 $对应于单线性方程的情况下解决$ 0 $陈述。此外，我们在（对称）随机RADO定理的0序列的原始证明中缩小了差距。
A famous result of Rado characterises those integer matrices $A$ which are partition regular, i.e. for which any finite colouring of the positive integers gives rise to a monochromatic solution to the equation $Ax=0$. Aigner-Horev and Person recently stated a conjecture on the probability threshold for the binomial random set $[n]_p$ having the asymmetric random Rado property: given partition regular matrices $A_1, \dots, A_r$ (for a fixed $r \geq 2$), however one $r$-colours $[n]_p$, there is always a colour $i \in [r]$ such that there is an $i$-coloured solution to $A_i x=0$. This generalises the symmetric case, which was resolved by Rödl and Ruciński, and Friedgut, Rödl and Schacht. Aigner-Horev and Person proved the $1$-statement of their asymmetric conjecture. In this paper, we resolve the $0$-statement in the case where the $A_i x=0$ correspond to single linear equations. Additionally we close a gap in the original proof of the 0-statement of the (symmetric) random Rado theorem.
