An asymptotic solution to the Erdős four-edge intersection problem

Abstract: For an $n$-vertex graph $G$ and a permutation $σ$ of its vertex set, let $σ(G)$ denote the corresponding relabelling of $G$, and put $I_G(σ)=|E(G)\cap E(σ(G))|$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(σ)\geq k$ for every $σ$. In his 1977 formulation of the problem, Erdős discussed the small values of $k$ and left the cases $k=4$ and $k=5$ as the next natural open questions. For $k=4$ he asked whether $f(n,4)=2n-4$, with the upper bound witnessed by $K_{2,n-2}$; the neighbouring $k=5$ question was recently settled exactly by Fang and Hou. We prove that every graph $G$ of order $n$ and size at most $2n-10n<sup>{2/3}-7$ has a relabelling with at most three common edges. Consequently, [ 2n-10n^{2/3}-7<f(n,4)\leq 2n-4, ] and hence [ f(n,4)=2n-o(n). ] Thus we resolve Erdős's four-edge intersection problem asymptotically, confirming his proposed value up to a sublinear error term. For comparison, for all sufficiently large $n$, Fang and Hou's result guarantees at most four common edges for graphs with at most $2n-3$ edges, whereas reducing the edge bound by only $10n<sup>{2/3}+4=o(n)$ already allows us to guarantee at most three common edges.

• The paper proves that for every n≥6, 2n−10n^{2/3}−7 < f(n,4) ≤ 2n−4, establishing the asymptotic formula f(n,4)=2n−o(n).
• The proof combines minimal-counterexample degree analysis, classical graph-packing theorems, Hall’s-theorem absorption, and a probabilistic pairing of tree components to construct relabellings with at most three shared edges.
• The result confirms Erdős’s conjecture up to an O(n^{2/3}) additive gap, while the exact value f(n,4)=2n−4 and extensions to larger intersection thresholds remain open.

The problem and its history

For an $n$-vertex graph $G$ and a permutation $\sigma$ of $V(G)$, the quantity $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$ counts the edges shared by $G$ and its relabelling $\sigma(G)$. Erdős's 1977 minimum-intersection problem asks, for each $k\geq1$, for the value of $f(n,k)$, the minimum size of an $n$-vertex graph such that every relabelling shares at least $G$0 edges with the original. Equivalently, every graph with fewer than $G$1 edges admits a relabelling sharing at most $G$2 edges. Erdős recorded $G$3 and $G$4 (citing Chung, Graham, and Murty), and observed that $G$5 gives $G$6 for $G$7. He conjectured equality — the Erdős four-edge intersection problem — and posed the analogous question $G$8 (Mullin's construction) alongside it. The five-edge case was recently settled exactly by Fang and Hou (Fang et al., 13 Aug 2026), who proved $G$9 for all sufficiently large $\sigma$0.

The paper works in the language of near-packings: an $\sigma$1-near-packing of $\sigma$2 is a permutation $\sigma$3 for which the common-edge graph $\sigma$4 belongs to $\sigma$5, where $\sigma$6 denotes the graphs with at most $\sigma$7 edges. The identity $\sigma$8, where $\sigma$9 is the corresponding near-packing threshold, reduces the four-edge problem to determining $V(G)$0.

Main result

The central theorem states that for every $V(G)$1,

$V(G)$2

and consequently $V(G)$3. The lower bound is the paper's contribution: every graph of order $V(G)$4 with at most $V(G)$5 edges has a relabelling with at most three common edges. This confirms Erdős's conjectured value $V(G)$6 up to a sublinear error term, resolving the four-edge problem asymptotically. The comparison with the five-edge result is instructive: Fang and Hou guarantee at most four common edges at edge bound $V(G)$7; reducing the budget by only $V(G)$8 edges already improves the guarantee to three common edges. The two results therefore sit at essentially the same linear edge scale $V(G)$9 while differing by one in the guaranteed intersection.

Structure of the proof

The proof is a minimal-counterexample argument with the target bound $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$0, where $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$1 and $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$2. Three standard ingredients are assembled: the Sauer–Spencer packing theorem (graphs $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$3 with $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$4 pack edge-disjointly in $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$5), the Bollobás–Eldridge/Burns–Schuster result that every graph of size at most $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$6 is packable, and a Hall's-theorem extension lemma stating that an independent set $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$7 of vertices of degree at most $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$8 with pairwise disjoint neighbourhoods and $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$9 can be absorbed into any near-packing of $G$0 without changing the set of common edges.

A degree-counting proposition ensures that whenever a vertex set $G$1 covers at least $G$2 edges, deleting $G$3 preserves the edge budget $G$4, so minimality of the counterexample yields an $G$5-near-packing of $G$6.

The structural core is a claim showing a minimal counterexample has no isolated vertices and at most seven leaves. Each case (isolated vertex; adjacent leaves; leaves sharing a neighbour; leaves with distinct neighbours) deletes a small vertex set covering at least twice its cardinality in edges and extends a near-packing of the remainder by a short cycle or transposition, using the extension lemma with $G$7 in the final case. The claim implies every vertex has degree at least two, with at most seven exceptions.

The global near-packing argument

With $G$8 and degrees at least $G$9 except for at most $\sigma(G)$0 vertices, a counting argument bounds the number $\sigma(G)$1 of high-degree vertices by $\sigma(G)$2. A maximal independent set $\sigma(G)$3 of vertices with degrees in $\sigma(G)$4 and pairwise disjoint neighbourhoods is nonempty but has $\sigma(G)$5 (otherwise the extension lemma would finish the proof), so its neighbourhood $\sigma(G)$6 satisfies $\sigma(G)$7.

The key structural lemma handles the "minimal components": trees $\sigma(G)$8 of $\sigma(G)$9 in which every vertex has at most one neighbour in $k\geq1$0. Pairing these trees arbitrarily, the lemma constructs a permutation moving every vertex of $k\geq1$1 outside $k\geq1$2 in both directions while creating at most three common edges. The construction is probabilistic in flavour: within each pair, a leaf $k\geq1$3 is extracted so that the remainder is packable by the $k\geq1$4 theorem; two random permutations $k\geq1$5 of $k\geq1$6 are then chosen so that the expected number of conflicting edge images is at most $k\geq1$7 and at most $k\geq1$8 respectively, giving a total of at most three common edges. This lemma is the technical heart of the paper and is what makes the $k\geq1$9 error term achievable.

The edge count then forces $f(n,k)$0 minimal components. Taking $f(n,k)$1 of them together with $f(n,k)$2 yields a subgraph $f(n,k)$3 to which the lemma applies, while the complement $f(n,k)$4 has fewer than $f(n,k)$5 edges and is therefore packable. Combining the two permutations, the property $f(n,k)$6 ensures no cross-edge between the parts can become common. This contradiction eliminates the counterexample and establishes the theorem.

Limitations and open questions

The result is asymptotic: the exact value of $f(n,k)$7 remains open, and Erdős's conjecture $f(n,k)$8 is confirmed only up to the additive error $f(n,k)$9. The gap between the upper and lower bounds is $n$0, so closing it to an exact determination would require either sharpening the near-packing threshold $n$1 or finding better obstructions than $n$2. The paper also leaves open the analogous question of whether the methods extend to $n$3, and the small-$n$4 regime ($n$5 is handled trivially by the $n$6 packing theorem) is not analyzed. The proof depends on the classical packing theorems of Sauer–Spencer and Bollobás–Eldridge, so any refinement of those results could potentially improve the error term.

Conclusion

The paper determines the asymptotic value of the Erdős four-edge intersection function, proving $n$7 via the explicit bound $n$8. Combined with Erdős's $n$9 construction, this confirms the conjectured value $G$00 up to a sublinear additive term, and places the $G$01 case on the same linear scale as the recently settled $G$02 case of Fang and Hou. The proof combines minimal-counterexample degree analysis, the Hall's-theorem absorption lemma, and a pairing argument over tree components of $G$03 governed by two random permutations, yielding a near-packing with at most three common edges. The exact value of $G$04, and the gap of order $G$05, remain open.