Erdős Problem #1178

Erdős Problem 1178 --- GPT 6 Astra Ultra partial attempt

FORMAL RESTATEMENT

For integers $r,e\geq3$, let $\mathcal F_{r,d,e}$ comprise the $r$-uniform hypergraphs having $d$ vertices and $e$ edges. Write $\operatorname{ex}_r(n,\mathcal F)$ for the largest number of edges in an $n$-vertex $r$-uniform hypergraph containing no member of $\mathcal F$ as a subhypergraph. Define \[d_r(e)=\min\{d:\operatorname{ex}_r(n,\mathcal F_{r,d,e})=o(n^2)\text{ as }n\to\infty\}.\] The Brown--Erdős--Sós conjecture asks whether $d_r(e)=(r-2)e+3$ for every $r,e\geq3$.

CONTEXT AND SCOPE

This replaces the earlier statement-only record with an elementary mathematical attempt. The arguments below establish only their stated partial results; no novelty or full solution is claimed.

ATTACK PLAN AND WORK

An elementary packing gives a useful lower obstruction. A linear $r$-uniform hypergraph is one in which two different edges meet in at most one vertex. A greedy packing of $r$-sets on $n$ vertices can select at least \[\frac{\binom nr}{\binom r2\binom{n-2}{r-2}} =\frac{\binom n2}{\binom r2^2}\] edges up to rounding: each selected edge rules out at most $\binom r2\binom{n-2}{r-2}$ candidates sharing a pair with it. Thus linear examples with $\gg_r n^2$ edges exist.

Any $e$ edges in a linear hypergraph have union size at least \[re-\binom e2.\tag{1}\] To verify this without an unjustified full inclusion-exclusion truncation, sum vertex multiplicities $m_v$. The overcount $\sum_v(m_v-1)$ is at most $\sum_v\binom{m_v}{2}$, which is at most $\binom e2$ by linearity. Hence, whenever the right side is positive, no collection of $e$ edges fits on fewer vertices, and \[d_r(e)\ge re-\binom e2.\tag{2}\] For $e=3$, this is exactly the conjectured necessary lower bound $3r-3$. It does not prove the matching upper bound. A crude general upper bound is $d_r(e)\le re$, because any $e$ edges occupy at most $re$ vertices; when $n\ge re$, a hypergraph avoiding every $re$-vertex $e$-edge configuration therefore has at most $e-1$ edges.

VERIFICATION AND REMAINING GAP

Prove the sharp subquadratic upper bound at $(r-2)e+3$ vertices and the matching lower obstruction for general $e$. The elementary linear packing only reaches the stated weaker range.

SOURCES

https://www.erdosproblems.com/1178

FINAL

UNRESOLVED. The full source problem remains outside the scope of the proved partial results.

COMPLETION ESTIMATE

$8\%$. This is a conservative subjective estimate toward the whole problem, not a probability that the argument is correct or a claim that a fixed fraction of the problem has been solved.