Erdős Problem 30: Multifaceted Challenges

• Erdős Problem 30 is a compilation-dependent designation that encapsulates multiple open questions across extremal graph theory, geometry, cycle-length phenomena, path covers, and additive number theory.
• It involves diverse formulations such as nonhamiltonian graph bounds, optimizing unit-distance edges, forcing distinct cycle lengths, and extremal thresholds in combinatorial and number-theoretic settings.
• The topic challenges researchers with rigorous stability analyses, combinatorial constructions, and analytic techniques that deepen understanding of extremal structures in graphs and arithmetic progressions.

“Erdős Problem 30” is not a uniquely fixed mathematical statement across the modern literature. In recent arXiv papers, the label is attached to several different questions attributed to Erdős, or to Erdős with coauthors, and these questions range over extremal graph theory, unit-distance geometry, cycle-length problems, path covers, additive number theory, and Dirichlet-series nonvanishing. This suggests that the designation is compilation-dependent: the same number is being inherited from different problem lists rather than from a single canonical statement (Füredi et al., 2016, Alexeev et al., 2024, Carr, 13 May 2026, Liu et al., 2020, Bedert, 2023, Yip, 18 Dec 2025).

1. Compilation-dependent designation

Several papers explicitly show that the label is list-specific. One paper treats the equal-degree endpoint question as “Erdős Problem 30 / Problem #816”; another treats the distinct-cycle-length question as “Erdős Problem 30 / Problem 11”; a third identifies the Erdős–Ingham nonvanishing question with “Erdős Problem 30 / Problem 967.” A plausible implication is that the phrase should always be read together with its surrounding source or statement, not as a self-sufficient identifier (Zhao et al., 5 May 2026, Lai, 2021, Yip, 18 Dec 2025).

Formulation of “Erdős Problem 30”	Representative statement	Paper
Nonhamiltonian graphs	extremal edge bounds and stability under a minimum-degree condition	(Füredi et al., 2016)
Unit-distance problem	maximize the number of unit-distance edges on $n$ vertices	(Alexeev et al., 2024)
Erdős–Gyárfás conjecture	$\delta(G)\ge 3$ should force a cycle of power-of-$2$ length	(Carr, 13 May 2026)
Odd cycle problem	odd cycle lengths in high-chromatic graphs have large reciprocal sum	(Liu et al., 2020)
Distinct cycle lengths	maximize edges when no two cycles have the same length	(Lai, 2021)
Monochromatic path covers	cover $V(K_n)$ by at most $\sqrt n$ same-colour paths	(Pokrovskiy et al., 2024)
Equal-degree odd-path problem	equal-degree vertices joined by a short odd path above an extremal threshold	(Zhao et al., 5 May 2026)
Property $P$	no $z<x,y$ in $A$ with $z\mid x+y$	(Bedert, 2023)
Erdős–Ingham question	nonvanishing of $1+\sum a_k^{-1-it}$	(Yip, 18 Dec 2025)

The breadth of this table is itself part of the subject. In current usage, “Erdős Problem 30” functions less as a theorem title than as a bibliographic marker whose meaning must be disambiguated locally.

2. Extremal graph-theoretic formulations

One graph-theoretic use of the label concerns Erdős’s theorem on nonhamiltonian graphs with prescribed minimum degree. For integers $\delta(G)\ge 3$0, the relevant quantities are

$\delta(G)\ge 3$1

Erdős proved that every nonhamiltonian $\delta(G)\ge 3$2-vertex graph $\delta(G)\ge 3$3 with $\delta(G)\ge 3$4 satisfies $\delta(G)\ge 3$5. The 2016 stability refinement identifies the near-extremal structure. It defines

$\delta(G)\ge 3$6

and proves that if $\delta(G)\ge 3$7, $\delta(G)\ge 3$8 is $\delta(G)\ge 3$9-connected and nonhamiltonian, $2$0, and $2$1, then $2$2 is a subgraph of $2$3 or $2$4. Here $2$5 is obtained from a clique $2$6 by adding $2$7 vertices of degree exactly $2$8, all adjacent to the same fixed $2$9 clique vertices, while $V(K_n)$0 is the edge-disjoint union of $V(K_n)$1 and $V(K_n)$2 sharing one vertex. The gap

$V(K_n)$3

is central to the stability interpretation, and for $V(K_n)$4 it is at least $V(K_n)$5 (Füredi et al., 2016).

A second extremal formulation asks: for which graphs $V(K_n)$6 does every graph on $V(K_n)$7 vertices and $V(K_n)$8 edges contain at least two copies of $V(K_n)$9? The answer is negative. For every integer $\sqrt n$0, there exists a graph $\sqrt n$1 of order $\sqrt n$2 and at least two values of $\sqrt n$3 such that some graph of order $\sqrt n$4 and size $\sqrt n$5 contains exactly one copy of $\sqrt n$6. The paper obtains such examples through stars $\sqrt n$7 and books $\sqrt n$8, and it also gives a detailed analysis of $\sqrt n$9: for every $P$0 with $P$1, there exists a graph of order $P$2 and size $P$3 containing exactly one copy of $P$4, whereas for $P$5 or $P$6 the minimum number of copies is $P$7 (Qiao et al., 2020).

These two uses share an extremal philosophy but ask different questions. In one case the issue is rigidity near the nonhamiltonian edge maximum; in the other it is the multiplicity of a forbidden subgraph immediately above the Turán threshold.

3. Cycle-length problems

Another family of uses of “Erdős Problem 30” is organized around cycle lengths. In the Erdős–Gyárfás conjecture, the statement is that every graph $P$8 with minimum degree $P$9 contains a cycle whose length is a power of $z<x,y$0. A 2026 paper studies minimal counterexamples and shows that every vertex of such a counterexample is adjacent to a vertex of degree exactly $z<x,y$1, the set of vertices of degree at least $z<x,y$2 forms an independent set, every regular minimal counterexample must be cubic, and at least $z<x,y$3 of the vertices have degree exactly $z<x,y$4. This is a structural narrowing of the search space rather than a resolution of the conjecture itself (Carr, 13 May 2026).

A different cycle-length problem, attributed in one paper to Erdős and Hajnal, asks whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. The solution is quantitative: if $z<x,y$5, then

$z<x,y$6

The paper further proves that for every $z<x,y$7, all sufficiently large $z<x,y$8-chromatic graphs contain all odd integers in an interval $z<x,y$9 as cycle lengths. In this formulation, “Problem 30” concerns the density of odd cycle lengths forced by large chromatic number (Liu et al., 2020).

A third cycle-length variant studies graphs in which any two cycles have different lengths. Let $A$0 be the maximum number of edges in a simple graph on $A$1 vertices with that property, let $A$2 be the extremal family, and let $A$3 be the maximum cycle length among graphs in $A$4. The main theorem states that for $A$5 sufficiently large,

$A$6

The same paper conjectures

$A$7

Here the “Problem 30” designation refers not to forcing a particular cycle length, but to the structure of extremal graphs in which cycle lengths are pairwise distinct (Lai, 2021).

Taken together, these examples show that cycle theory alone does not isolate a single Problem 30. The label has been attached to at least three distinct cycle-length phenomena: power-of-$A$8 cycles under a minimum-degree hypothesis, odd-cycle-length richness under a chromatic hypothesis, and extremal sparsity of repeated cycle lengths.

4. The unit-distance formulation

In geometric extremal combinatorics, “Erdős Problem 30” has been used for the finite-$A$9 unit-distance question. A unit-distance graph is a simple graph $z\mid x+y$0 admitting an injective map $z\mid x+y$1 such that $z\mid x+y$2 implies $z\mid x+y$3. Writing $z\mid x+y$4 for the set of unit-distance graphs on $z\mid x+y$5 vertices,

$z\mid x+y$6

is the extremal quantity. A 2024 paper proves the exact values

$z\mid x+y$7

improves the upper bounds for $z\mid x+y$8 to

$z\mid x+y$9

$1+\sum a_k^{-1-it}$0

and records the lower bounds

$1+\sum a_k^{-1-it}$1

$1+\sum a_k^{-1-it}$2

Thus for $1+\sum a_k^{-1-it}$3 the paper establishes

$1+\sum a_k^{-1-it}$4

It also fully enumerates the densest unit-distance graphs for every $1+\sum a_k^{-1-it}$5 (Alexeev et al., 2024).

The methodology is three-layered. First, the paper uses a set $1+\sum a_k^{-1-it}$6 of $1+\sum a_k^{-1-it}$7 minimal forbidden subgraphs for unit-distance graphs on at most $1+\sum a_k^{-1-it}$8 vertices and defines $1+\sum a_k^{-1-it}$9 as the set of $\delta(G)\ge 3$00-free simple graphs on $\delta(G)\ge 3$01 vertices, with

$\delta(G)\ge 3$02

Second, it deploys “totally unfaithful” unit-distance graphs to eliminate dense candidates that would force additional unit edges in every embedding. Third, it uses a custom embeddability solver in $\delta(G)\ge 3$03, with logic moves labeled $\delta(G)\ge 3$04–$\delta(G)\ge 3$05, designed to avoid general cylindrical algebraic decomposition. In this setting, “Erdős Problem 30” is the graph-theoretic finite reformulation of the classical planar unit-distance problem (Alexeev et al., 2024).

5. Path-cover and equal-degree path formulations

One asymptotic formulation concerns monochromatic path covers in $\delta(G)\ge 3$06-edge-coloured complete graphs. Erdős and Gyárfás proved in 1995 that every $\delta(G)\ge 3$07-edge-coloured complete graph on $\delta(G)\ge 3$08 vertices has a collection of $\delta(G)\ge 3$09 monochromatic paths, all of the same colour, covering the entire vertex set, and they conjectured that $\delta(G)\ge 3$10 should suffice. The asymptotic form has now been proved: there exists $\delta(G)\ge 3$11 such that for all $\delta(G)\ge 3$12 and all $\delta(G)\ge 3$13-edge-colourings of $\delta(G)\ge 3$14, there exists a collection of at most $\delta(G)\ge 3$15 monochromatic paths of the same colour that cover the vertices of $\delta(G)\ge 3$16. The same paper notes that, with extra technical work, $\delta(G)\ge 3$17 monochromatic paths of the same colour suffice for all $\delta(G)\ge 3$18. In its conventions, paths may have length zero and need not be disjoint (Pokrovskiy et al., 2024).

Another path-based formulation begins with a 1991 question of Erdős and Hajnal: is it true that every graph on $\delta(G)\ge 3$19 vertices with $\delta(G)\ge 3$20 edges contains two vertices of equal degree joined by a path of length three? The extremal example is $\delta(G)\ge 3$21, which has exactly $\delta(G)\ge 3$22 edges and does not contain such a pair, so the threshold is sharp if the statement holds. A 2026 paper confirms the fixed-odd-length generalization conjectured by Chen and Ma: for every fixed $\delta(G)\ge 3$23, if $\delta(G)\ge 3$24 is a graph on $\delta(G)\ge 3$25 vertices with at least $\delta(G)\ge 3$26 edges and no two vertices of equal degree are joined by a path of length $\delta(G)\ge 3$27, then for all sufficiently large $\delta(G)\ge 3$28,

$\delta(G)\ge 3$29

In this use of the label, “Problem 30” refers to an extremal threshold for forcing equal-degree endpoints on a short odd path (Zhao et al., 5 May 2026).

These two path formulations are distinct. One asks for global covering by few monochromatic paths in a complete graph; the other asks for the unavoidable appearance of an equal-degree pair linked by a prescribed odd path length above a bipartite extremal threshold.

6. Additive, multiplicative, and analytic number-theoretic formulations

In additive combinatorics, one supplied source identifies the Erdős conjecture on arithmetic progressions as the closest match to an “Erdős Problem 30” in that area, while also noting that the survey paper does not explicitly use the number. The conjecture is: $\delta(G)\ge 3$30 The same source stresses that this remains open, and that even the $\delta(G)\ge 3$31-term case is unknown in this generality. It also gives the finite extremal reformulation asking how large a subset $\delta(G)\ge 3$32 must be to guarantee a $\delta(G)\ge 3$33-term arithmetic progression, with the “suggested answer” being around $\delta(G)\ge 3$34 (Gowers, 2015).

A different additive-divisibility formulation is the Erdős–Sárközy property $\delta(G)\ge 3$35. A set $\delta(G)\ge 3$36 has property $\delta(G)\ge 3$37 if there are no $\delta(G)\ge 3$38 with

$\delta(G)\ge 3$39

The finite extremal question asks whether every $\delta(G)\ge 3$40 with property $\delta(G)\ge 3$41 satisfies $\delta(G)\ge 3$42. This has now been resolved asymptotically: for all sufficiently large $\delta(G)\ge 3$43,

$\delta(G)\ge 3$44

and the paper also proves the global bound

$\delta(G)\ge 3$45

for an absolute constant $\delta(G)\ge 3$46. The sharp example is the top third

$\delta(G)\ge 3$47

which has size $\delta(G)\ge 3$48 (Bedert, 2023).

The multiplicative generalization uses property $\delta(G)\ge 3$49: a set $\delta(G)\ge 3$50 has $\delta(G)\ge 3$51 if there do not exist distinct elements

$\delta(G)\ge 3$52

such that

$\delta(G)\ge 3$53

The counting problem asks for

$\delta(G)\ge 3$54

For $\delta(G)\ge 3$55, the number of such subsets is

$\delta(G)\ge 3$56

while for every fixed $\delta(G)\ge 3$57,

$\delta(G)\ge 3$58

Here

$\delta(G)\ge 3$59

and

$\delta(G)\ge 3$60

(Pach et al., 2020).

An analytic-number-theoretic variant is the Erdős–Ingham question. Given

$\delta(G)\ge 3$61

Erdős and Ingham asked whether for every real $\delta(G)\ge 3$62,

$\delta(G)\ge 3$63

The infinite-sequence version is false in a strong sense: for any complex number $\delta(G)\ge 3$64 and any non-zero real $\delta(G)\ge 3$65, there exists a sequence $\delta(G)\ge 3$66 such that

$\delta(G)\ge 3$67

Taking $\delta(G)\ge 3$68 yields a direct counterexample to the nonvanishing statement. The same paper notes that the finite-set variant remains open, and specifically points out that even the case $\delta(G)\ge 3$69 is unresolved (Yip, 18 Dec 2025).

Across these number-theoretic examples, the recurring theme is again bibliographic rather than thematic unity. “Erdős Problem 30” may refer to a borderline-density arithmetic-progression conjecture, a divisibility-avoidance extremal problem, a counting problem for multiplicative configurations, or a Dirichlet-series nonvanishing question, depending on the source tradition being followed.