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Erdős assigns signs to edges $e(i,j)$ on printed p.30 / PDF p.2 of the archive's scan (https://users.renyi.hu/~p_erdos/1963-15.pdf). The sum on a vertex subset counts each unordered edge once. On printed p.31 / PDF p.3, he defines $H(n)$ as the minimum over edge colorings of the largest absolute sum on an induced complete subgraph. Theorem II then displays

$$\frac n4\le H(n)<C_4n^{3/2}.$$

This is the normalized quantity in the convention record. The displayed theorem does not state a small-$n$ threshold. It cannot literally cover $n=1$, where there are no edges and $H(1)=0$. We retain it as the historical linear lower and order-$n^{3/2}$ upper bound without inferring all-$n$ endpoints from the omitted range. The later Erdős--Spencer theorem states its sufficiently-large-$n$ range explicitly.

The constant $C_4$ is not given a value; the paper's convention on p.29 makes $c_1,c_2,\ldots$ positive constants, and its proof takes $C_4$ to be a sufficiently large absolute constant. The English summary on printed p.37 restates the theorem as $n/4<H(n)<cn^{3/2}$, with a strict lower inequality. The paper suggests (p.31) that the upper bound is probably very poor and that the lower bound may not be far from the true order, though $\frac14$ can surely be replaced by a larger value.

Proof pointer. Lower bound, printed p.35: after exchanging the two classes if needed, a vertex has at least $[(n-1)/2]$ edges in the first class; either those neighbours span a sum at most $-n/4$, or adding the vertex gives a sum the paper concludes is at least $n/4$. As printed, the second case adds $[(n-1)/2]$ to a sum exceeding $-n/4$, which gives more than $n/4-1$; this page does not reconstruct how the stated $n/4$ follows for every $n$. Upper bound, printed p.36, not given in detail: as in (17)--(18) for Theorem I, for each fixed complete subgraph fewer than $2^{\binom n2-n}$ signings give it a sum of absolute value at least $C_4n^{3/2}$, and a union bound over the $2^n$ vertex subsets leaves a signing with every such sum at most $C_4n^{3/2}$.

Proof scope. The definition, the statement and the proof on pp.35--36 were read on the printed pages. The tail count behind the upper bound is the one the paper leaves to the reader for (18), and was not re-derived; no arbitrary ordered-pair assertion is made here.

Source. P. Erdős, Ramsey és Van der Waerden tételével kapcsolatos kombinatorikai kérdésekről, Mat. Lapok 14 (1963), 29--37, Theorem II, printed p.31 / PDF p.3; proof pp.35--36.

Bears on. Problem 1028: the theorem bounds the problem's edge quantity, read with one sign per unordered edge, by $n/4\le H(n)<C_4n^{3/2}$ as printed; its upper bound has the order $n^{3/2}$ that the later Erdős--Spencer theorem shows is the true order for sufficiently large $n$, and its linear lower bound is not of that order.