One-way randomized communication complexity of Greater-Than Let $\mathrm{GT}_n:\{0,1\}^n \times \{0,1\}^n \to \{0,1\}$ be the greater than function: $\mathrm{GT}_n(x,y)=1$ exactly when the positive integer whose binary representation is $x$ is greater than the positive integer whose binary representation is $y$.
It is well-known that the randomized communication complexity of $\mathrm{GT}_n$ is $\Theta(\log n)$. Several references claim that the randomized one-way communication complexity of $\mathrm{GT}_n$ is $\Theta(n)$, but I am unable to find a reference that actually contains a proof. It is easy to show a $\Omega(n/\log n)$ lower bound by a reduction from INDEX, and this is done in a paper by Kremer, Nisan, and Ron. They also claim that the proof of the $\Omega(n)$ lower bound appears in a paper by Miltersen, Nisan, Safra, and Wigderson. The latter paper uses a round-elimination lemma to show a $\frac{n^{1/k}}{120^k}$ lower bound on the $k$-round communication complexity of $\mathrm{GT}_n$ but never gets around to proving the base case somehow (see Theorem 14), and the base case is all I am asking for. They also claim this is an unpublished result of Yao, but as far as I know that result may never have been written down.
Surely at some point someone wrote down a proof of this lower bound? The closest I could find (with help from Clément Canonne) is this paper showing a lower bound on the quantum one-way communication complexity of $\mathrm{GT}_n$ and that just seems like an overkill.
If you look at the MNSW proof carefully, the base case can be taken to be the trivial fact that a $0$-round protocol for $\textrm{GT}_n$ with $n = 1$ requires one bit of communication.
If the goal is simply to convince oneself from first principles that $\textrm{R}^\to(\textrm{GT}_n) = \Omega(n)$, this is immediate by reduction from the AUGMENTED-INDEX problem. An $\Omega(n)$ one-way lower bound for AUG-INDEX boils down to the chain rule for mutual information. For a discussion of AUG-INDEX, see, e.g., Section 3.4 of Jelani Nelson's PhD thesis (https://people.eecs.berkeley.edu/~minilek/publications/papers/phd_thesis.pdf).