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Statement. The printed theorem reads: "Consider a set $\mathcal P$ of $m$ points and a set $\mathcal Q$ of $n$ points, with $2 \leqslant m \leqslant n^{1/3}$. Then there exists a point in $\mathcal P$ that determines $\Omega(\sqrt{mn})$ distances with the points in $\mathcal Q$." (p. 7).

The sets lie in the plane. In the paper's notation $D(p,\mathcal Q)$ is the number of distinct distances from $p$ to the points of $\mathcal Q$, and the theorem says that $\max_{p\in\mathcal P}D(p,\mathcal Q)\geq c\sqrt{mn}$ for an absolute constant $c>0$, throughout the stated range. It implies Theorem 4, since $D(\mathcal P,\mathcal Q)\geq D(p,\mathcal Q)$ for every $p\in\mathcal P$ (p. 9). Remark 18 (p. 9) notes that in Elekes's construction, Proposition 6, every point of $\mathcal P$ determines $\Theta(\sqrt{mn})$ distances.

Source. Surya Mathialagan, On Bipartite Distinct Distances in the Plane, Electronic Journal of Combinatorics 28(4) (2021), P4.33, DOI 10.37236/9687: Theorem 14 on p. 7, proof on pp. 7--8, Proposition 15 on p. 8 with its proof on p. 9. The copy read is identified on the source card.

Proof sketch. This is Székely's crossing-number method adapted to two sets. Let $t$ be the largest number of distances from a point of $\mathcal P$ to $\mathcal Q$, and suppose $t\leq\epsilon\sqrt{mn}$ for a small constant $\epsilon$. Around each $p\in\mathcal P$ draw the at most $t$ circles centred at $p$ that pass through points of $\mathcal Q$. Join consecutive points of $\mathcal Q$ along each circle, and discard the circles carrying at most two points. This leaves a multigraph on the $n$ points with $\Theta(mn)$ edges, drawn with $O(m^2t^2)$ crossings, since the at most $mt$ circles meet pairwise in at most two points. Many parallel edges between $u$ and $v$ force many centres of $\mathcal P$ onto the perpendicular bisector of $u$ and $v$, and Proposition 15 bounds how many edges such rich bisectors carry. Deleting the edges of multiplicity at least a large constant $K$ removes at most half of them, using $m^3\leq n$. Székely's crossing lemma for multigraphs (Theorem 11, p. 7) then gives $m^2t^2\gtrsim m^3n$, so $t\gtrsim\sqrt{mn}$.

Proposition 15 (p. 8). For an integer $r\geq2$, let $T$ be the set of pairs $(\ell,e)$ with $e=(u,v,C)$ an edge of the multigraph, $\ell$ the perpendicular bisector of $u$ and $v$, and $\ell$ incident to at least $r$ points of $\mathcal P$. Then $|T|=O(tm^2/r^2+tm\log m)$. Its proof (p. 9) combines the bound $O(m^2/r^3+m/r)$ on $r$-rich lines (Theorem 13, p. 7, from Szemerédi--Trotter) with a dyadic decomposition over $r$.

A range note. The crossing lemma for multigraphs is applied under its hypothesis $e>5cn$, here with multiplicity bound $K$. With $\Theta(mn)$ edges this holds once $m$ exceeds a constant depending on $K$; the proof does not treat smaller $m$ separately. For $m$ below any fixed bound the conclusion follows from two points of $\mathcal P$ alone. The points of $\mathcal Q$ lie on at most $t$ circles about each of two distinct centres, and two circles with distinct centres share at most two points, so $n\leq2t^2$ and $t\geq\sqrt{n/2}$. This note is this page's observation, not the paper's.

Dependencies. Theorem 11 (Székely's crossing lemma for multigraphs, p. 7) and Theorem 13 (the bound on $r$-rich lines, p. 7) are cited by the paper and not proved there. Remark 16 (p. 8) records that the restriction $m=O(n^{1/3})$ is used in the multiplicity step and that the same argument gives only $\Omega(m^{3/5}n^{1/5})$ for larger $m$. Remark 17 (p. 9) treats the case where $\mathcal P$ lies on a line, for every $2\leq m\leq n$.

Read depth. Claims checked: the statement and Proposition 15 were read clause by clause on the published PDF, and the proof of the theorem was followed step by step; the proof of Proposition 15 and the cited Theorems 11 and 13 were not re-derived. Nothing here is independently reviewed, and this page is outside the reviewed Theorem 3 record on this card.

Bears on. Problem 652: the claim page Mathialagan's point with many bipartite distances deduces the problem's answer from this theorem, applied to the $k$ points with fewest distances and the remaining points; that deduction and its standing are recorded there, not here.