For an integer (n\ge2), define
[
U_n=(n,2n)\cap\mathbb N
={n+1,n+2,\ldots,2n-1}
]
and
[
V_n=(2n,4n)\cap\mathbb N
={2n+1,2n+2,\ldots,4n-1}.
]

For a set (B\subseteq V_n), call a set (C\subseteq U_n) (B)-avoiding if
[
c_1+c_2\notin B
]
for every pair of distinct elements (c_1,c_2\in C).

Let (f(n)) be the largest integer (t) such that, for every set
[
B\subseteq V_n,
]
there exists a (B)-avoiding set
[
C\subseteq U_n
]
satisfying
[
|B|+|C|\ge t.
]

Equivalently, for each (B\subseteq V_n), define
[
\alpha_n(B)
===========

\max\left{
|C|:
C\subseteq U_n,\
c_1+c_2\notin B
\text{ for all distinct }c_1,c_2\in C
\right}.
]
Then
[
f(n)=\min_{B\subseteq V_n}\bigl(|B|+\alpha_n(B)\bigr).
]

Resolve the following Erdős problem completely:

Estimate the asymptotic growth of (f(n)). In particular, determine whether
[
f(n)\le n^{1/2+o(1)}
\qquad\text{as }n\to\infty.
]

The statement
[
f(n)\le n^{1/2+o(1)}
]
means precisely that, for every fixed (\varepsilon>0), there exists (n_0(\varepsilon)) such that
[
f(n)\le n^{1/2+\varepsilon}
]
for every (n\ge n_0(\varepsilon)).

Do not assume in advance that the conjectured upper bound is true or false. Assume for purposes of this task that a complete resolution of the principal question exists. A complete resolution must prove exactly one of the following two statements.

Affirmative resolution:

For every (\varepsilon>0), there exists (n_0(\varepsilon)) such that, for every (n\ge n_0(\varepsilon)), one can choose a set
[
B\subseteq V_n
]
for which every (B)-avoiding set (C\subseteq U_n) satisfies
[
|B|+|C|\le n^{1/2+\varepsilon}.
]

Equivalently,
[
f(n)\le n^{1/2+o(1)}.
]

Negative resolution:

There exists a fixed constant (\varepsilon>0) and infinitely many integers (n) such that, for every set
[
B\subseteq V_n,
]
there exists a (B)-avoiding set
[
C\subseteq U_n
]
satisfying
[
|B|+|C|>n^{1/2+\varepsilon}.
]

Equivalently,
[
f(n)>n^{1/2+\varepsilon}
]
for infinitely many (n).

The quantifier order is essential. To prove the conjectured upper bound, it is enough to construct, for each sufficiently large (n), one suitable set (B) for which every admissible (C) is small. To disprove it, one must prove that for infinitely many (n), every possible choice of (B) admits a sufficiently large (B)-avoiding set (C).

The solution should also obtain the strongest rigorously justified upper and lower bounds for (f(n)) that follow from its method, and should determine the correct asymptotic order if possible. However, the non-negotiable requirement is to settle the stated bound
[
f(n)\le n^{1/2+o(1)}
]
exactly. Do not claim to have fully estimated (f(n)) unless the proof actually determines its asymptotic order.

The problem has the following exact graph formulation. Given (B\subseteq V_n), define a simple graph (G_B) with vertex set (U_n), where distinct vertices (x,y\in U_n) are adjacent if and only if
[
x+y\in B.
]
Then a set (C\subseteq U_n) is (B)-avoiding if and only if it is an independent set in (G_B). Therefore
[
\alpha_n(B)=\alpha(G_B)
]
and
[
f(n)=\min_{B\subseteq V_n}
\bigl(|B|+\alpha(G_B)\bigr).
]

Each sum (s\in B) contributes the matching
[
\bigl{{x,y}\subseteq U_n:x\ne y,\ x+y=s\bigr}
]
to (G_B). Thus (G_B) is a union of sum matchings generated by the selected values (s\in B). This structure must be preserved in any graph-theoretic reformulation: (G_B) is not an arbitrary graph.

The restriction (c_1\ne c_2) is also essential. The condition does not forbid an element (c\in C) merely because
[
2c\in B.
]
The graph (G_B) is simple and has no loops.

Partial progress does not count unless it implies exactly one of the two resolutions above. In particular, the following are insufficient:

* proving only a bound of the form
  [
  f(n)\le n^{1/2+\delta}
  ]
  for one fixed (\delta>0);

* proving only
  [
  f(n)\le n^{1/2}(\log n)^C
  ]
  without explaining that the logarithmic factor is (n^{o(1)}) and controlling all other factors uniformly;

* proving the desired upper bound only along a subsequence of (n);

* constructing one set (B) for one large finite value of (n);

* proving computationally that the bound holds through any fixed range of (n);

* proving an upper bound for random (B) only with positive probability that tends to zero;

* computing only the expected independence number of (G_B) without proving the existence of a suitable deterministic (B);

* treating (G_B) as an arbitrary Erdős-Rényi random graph and ignoring its sum-matching dependencies;

* replacing the interval (U_n=(n,2n)\cap\mathbb N) or (V_n=(2n,4n)\cap\mathbb N) by cyclic groups, symmetric intervals, or different ranges without rigorously transferring the result back to the original problem;

* allowing (c_1=c_2), or imposing the stronger but different restriction (2c\notin B);

* proving only that (C+C) avoids (B), when this includes diagonal sums and is therefore stronger than the required condition, without showing that the stronger problem has the same asymptotic answer;

* finding a large independent set in one particular graph (G_B), since an upper bound for (f(n)) requires a graph with small independence number;

* constructing graphs with small independence number that cannot be represented as (G_B) for any (B\subseteq V_n);

* using an unproved conjecture about random Cayley graphs, random sum graphs, or their independence numbers;

* reducing the problem to another unproved threshold or extremal statement of comparable strength.

Standard proved theorems from additive combinatorics, random graph theory, random Cayley or sum graphs, extremal graph theory, probabilistic combinatorics, additive energy, Fourier analysis, entropy methods, container theory, large deviations, or optimization may be used, but they must be stated accurately and applied with all necessary hypotheses, dependencies, and uniformity.

Use multiagent v2 aggressively and dynamically. You have up to 4 concurrent agents available. Do not use a fixed assignment such as “N agents for strategy X.” Instead, manage the search using the following heuristics:

* Begin with a genuinely diverse portfolio of approaches. Agents should explore substantially different formulations, invariants, reductions, random and pseudorandom choices of (B), structured sum graphs, Cayley-graph analogues, independence-number bounds, additive-energy methods, Fourier analysis, entropy arguments, container methods, dependent random choice, alteration arguments, explicit constructions, algebraic constructions, interval-to-group transference, optimization, and computational sanity checks.

* Do not tell most agents the currently favored approach. Preserve independence during early rounds so that agents do not all converge to the same attractive but incomplete random-graph or random-Cayley heuristic.

* Maintain an explicit registry of approach families. Group agents by the mathematical idea they are using, not by superficial wording. If many agents converge to one family, redirect some of them toward underexplored formulations.

* Do not allow one approach to dominate merely because it gives an elegant reduction or the conjectured exponent heuristically. A route that ends at an unproved independence-number estimate equivalent in strength to the original conjecture is not close to completion unless it supplies a genuinely new proof of that estimate.

* When an approach stalls at a theorem-strength missing lemma, mark that route as blocked. Only continue assigning agents to it if someone proposes a materially new mechanism, invariant, construction, inequality, transference principle, or probabilistic argument.

* Keep several incompatible proof routes alive through multiple rounds. Maintain both conjectured-upper-bound routes and lower-bound or counterexample routes until one side is rigorously ruled out. Cross-pollinate ideas only after independent agents have developed them far enough to expose their real strengths and gaps.

* Use computational agents throughout. They should compute or approximate
  [
  \min_{B\subseteq V_n}\bigl(|B|+\alpha(G_B)\bigr)
  ]
  for small and medium (n), search for extremal choices of (B), optimize random and structured constructions, test proposed independence bounds, identify additive patterns, and find counterexamples to intermediate lemmas. Computation is evidence unless it is converted into a rigorous asymptotic argument or an exact finite certificate completing a valid reduction.

* Use adversarial agents throughout. Every candidate proof must be checked for:

  * the exact order of the quantifiers over (B) and (C);
  * the fact that (B) is chosen first and (C) may depend on (B);
  * the open integer intervals ((n,2n)) and ((2n,4n));
  * the requirement (c_1\ne c_2);
  * the absence of loop constraints from sums (2c);
  * the distinction between constructing a good (B) and proving a statement for every (B);
  * the distinction between expected independence number and high-probability or deterministic existence;
  * dependencies among edges generated by the same sum and among overlapping sum matchings;
  * unjustified replacement of the interval model by a cyclic or group model;
  * boundary losses in any transference argument;
  * logarithmic or subpolynomial factors hidden inside (n^{o(1)});
  * estimates proved only along a subsequence;
  * a construction of an arbitrary graph rather than a realizable sum graph (G_B);
  * circular use of an independence-number conjecture equivalent to the desired bound.

* Require agents to return concrete lemmas, constructions, probability estimates, equations, algorithms, code outputs, exact finite examples, certificates, or counterexamples to proposed sublemmas. Reject status reports, vague optimism, and claims that an unproved independence-number, pseudorandomness, or transference statement is “routine.”

* The root agent should repeatedly synthesize, challenge, redirect, and launch new rounds. Do not stop after the first wave fails. Produce a complete proof or a complete disproof only if it survives adversarial audit; otherwise report only the strongest rigorously proved derivation and its exact remaining gap.

Do not return merely because current approaches fail or agents report theorem-strength gaps. Continue launching new rounds, reopening blocked approaches only when there is a genuinely new mechanism, and searching for fresh formulations.

Return only when the bound
[
f(n)\le n^{1/2+o(1)}
]
has been completely resolved and the argument survives adversarial audit. Do not return a reduction, partial result, isolated missing lemma, finite computation, heuristic random-graph calculation, numerical guess, “best effort” summary, or explanation of why the problem is difficult.

Do not stop, return, or give up until you have found a complete and rigorous solution. Continue exploring every plausible approach, repairing failed arguments, and developing new ones until the problem is fully resolved.

Public search may be used only for ordinary mathematical background or standard named theorems, not to search for a solution to this exact Erdős problem or benchmark. Do not search the public web merely to determine whether this problem is open, and do not answer that it is open.