Browse Problems

If `f(z) = ∑ aₖzⁿₖ` is an entire function (with `aₖ ≠ 0` for all `k`) such that `∑ 1 / nₖ < ∞`, then `f` assumes every value infinitely often. This …

Let $f$ be a Rademacher multiplicative function. Does there exist some constant $c > 0$ such that, almost surely, \[ \limsup_{N \to \infty} \frac{\sum_{m \leq N} f(m)}{\sqrt{N \log \log N}} = c? …

Let `f(z)=∑_{0≤k≤n} ϵ_k z^k` be a random polynomial, where `ϵ_k∈{−1,1}` independently uniformly at random for `0≤k≤n`. Is it true that the number of roots of `f(z)` in `{z∈C:|z|≤1}` is, almost surely, `(1/2+o(1))n`? …

Erdős and Offord showed that the number of real roots of a random degree `n` polynomial with `±1` coefficients is `(2/π+o(1))log n`.

Yakir proved that almost all Kac polynomials have `n/2+O(n^(9/10))` many roots in `{z∈C:|z|≤1}`.

Let $\epsilon>0$ and $N$ be sufficiently large. Is it true that if $A\subseteq \{1,\ldots,N\}$ has size at least $\epsilon N$ then there must be distinct $a,b,c\in A$ such that $$[a, b]=[b, c]=[a, …

Let $a_1, \dots, a_p$ be (not necessarily distinct) residues modulo a prime $p$, such that there exists some $r$ so that if $S \subseteq [p]$ is non-empty and $$\sum_{i \in S} a_i …

Gao, Hamidoune, and Wang [GHW10] solved this for all moduli `p` (not necessarily prime). [GHW10] Gao, Weidong and Hamidoune, Yahya Ould and Wang, Guoqing, Distinct length modular zero-sum subsequences: a proof of …

This was proved by Erdős and Szemerédi [ErSz76] for p sufficiently large. [ErSz76] Erd\H os, E. and Szemerédi, E., On a problem of Graham. Publ. Math. Debrecen (1976), 123--127.

Suppose $A \subseteq \{1,\dots,N\}$ is such that there are no $k+1$ elements of $A$ which are relatively prime. An example is the set of all multiples of the first $k$ primes. Is …

Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some monochromatic …

Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform hypergraph on …

Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then $\hat{r}(G,H) \ll m$? …

**Erdős Problem 567 (H5)** Is $H_5$ ($C_5$ with two vertex-disjoint chords) Ramsey size linear?

**Erdős Problem 567 (K33)** Is $K_{3,3}$ Ramsey size linear?

**Erdős Problem 567 (Q3)** Is $Q_3$ (the 3-dimensional hypercube) Ramsey size linear?

Nguyen and Vu proved that $|A| \ll N^{1/3} (\log N)^{O(1)}$.

Let $α$ be the infinite ordinal $\omega^{\omega}$. It was proved by Chang [Ch72] that any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.

Let m be a finite cardinal $< \omega$. Let $α$ be the infinite ordinal $\omega^{\omega}$. It was proved by Milnor that any red/blue colouring of the edges of $K_α$ there is either …

Specker [Sp57] proved that when $α=ω^n$ for $3≤ n < \omega$ then it is not the case that any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ …

Specker [Sp57] proved that when $α=ω^2$ any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$.

Let $α$ be the infinite ordinal $\omega^{\omega^2}$. Is it true that any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$? This is true …

Determine which countable ordinals $β$ have the property that, if $α = \omega^β$, then in any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue …

**Erdős Problem 598:** Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$. Can one colour the countable subsets of $m$ using $\kappa$ many colours so that every …

There are infinitely many $n$ such that $d_n < d_{n+1} < d_{n+2}$, where $d$ denotes the prime gap function.

The Erdős–Hajnal Conjecture states that there is a constant $c(H) > 0$ for each $H$ such that we can take $f(n) = n^{c(H)}$ in the above formulation.

Let $r\geq 3$. If the edges of $K_{r^2+1}$ are $r$-coloured then there exist $r+1$ vertices with at least one colour missing on the edges of the induced $K_{r+1}$. In other words, there …

Erdős and Gyárfás [ErGy99] showed this property fails for infinitely many $r$ if we replace $r^2+1$ by $r^2$.

Erdős and Gyárfás [ErGy99] proved the conjecture for $r=3$.

Erdős and Gyárfás [ErGy99] proved the conjecture for $r=4$.

Bucić, Nguyen, Scott, and Seymour [BNSS23] improved this to $f(n) = \exp(c_H \sqrt{\log n \log \log n})$ for some constant $c_H > 0$ dependending on $H$. [BNSS23] Bucić, M. and Nguyen, T. …

Erdős and Hajnal [ErHa89] proved that we can take $f(n) = \exp(c_H \sqrt{\log n})$ for some constant $c_H > 0$ dependending on $H$. [ErHa89] Erdős, P. and Hajnal, A., Ramsey-type theorems. Discrete …

Let $X$ be a set of cardinality $\aleph_\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\not\in A$ for all $A$. Must there exist an …

Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:\{A : A\subseteq X\}\to X$ so that for every $Y\subseteq X$ with $\lvert Y\rvert …

Does every finite graph with minimum degree at least $3$ contain a cycle of length $2^k$ for some $k \geq 2$?

If ℕ is $2$-coloured then there must exist a monochromatic three-term arithmetic progression $x,x+d,x+2d$ such that $d>x$.

Let $\tau(n)$ count the number of divisors of $n$. Is there some $n > 24$ such that $$ \max_{m < n}(m + \tau(m)) \leq n + 2? $$

Erdős says it 'seems certain' that for every $k$ there are infinitely many $n$ for which $$ \max_{n−k < m < n}(m + \tau(m)) ≤ n + 2. $$

Erdős says 'it is extremely doubtful' that there are infinitely many such $n$, and in fact suggests that $$ lim_{n\to\infty} \max_{m < n}(\tau(m) + m − n) = \infty. $$

This is true for $n = 24$.

Is there a set of $n$ points in $\mathbb{R}^2$ such that every subset of $4$ points determines at least $3$ distances, yet the total number of distinct distances is $\ll \frac{n}{\sqrt{\log n}}$? …

Is there and $A \subset \mathbb{N}$ is such that $$\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}$$ exists and is $\ne 0$?

**The Erdős discrepancy problem** If $f\colon \mathbb N \rightarrow \{-1, +1\}$ then is it true that for every $C>0$ there exist $d, m \ge 1$ such that $$\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert > …

Can the product of an arithmetic progression of positive integers $n, n + d, ..., n + (k - 1)d$ of length ≥ 4, with $(n, d) = 1$, be a perfect …

According to https://www.erdosproblems.com/672, Euler proved this.

According to https://www.erdosproblems.com/672, Obláth proved this. [Ob51] Oblath, Richard, Eine Bemerkung über Produkte aufeinander folgender Zahlen. J. Indian Math. Soc. (N.S.) (1951), 135-139.

Denote by $M(n, k)$ the least common multiple of the finite set $\{n+1, \dotsc, n+k\}$. Is it true that for all $m \geq n + k$, we get $M(m, k) \neq M(n, …

Write $M(n, k)$ be the least common multiple of $\{n+1, \dotsc, n+k\}$. Let $k$ be sufficiently large. Are there infinitely many $m, n$ with $m \geq n + k$ such that $$ …

**The Erdős discrepancy problem (complex variant)** If $f\colon \mathbb N \rightarrow S^1 ⊆ ℂ$ then is it true that for every $C>0$ there exist $d, m \ge 1$ such that $$\left\lvert \sum_{1\leq …

Is $$\sum_{n=2}^\infty \frac{1}{n!-1}$$ irrational?