Erdős and Rado proved that $\mathfrak{c}\to (\omega+n,4)_2^3$ for any $2\leq n<\omega$.

A problem of Erdős, Hajnal, and Szemerédi [EHS82]. Every $G$ with chromatic number $\aleph_1$ must have $h_G(n)\gg n$ since $G$ must contain, for some $r$, $\aleph_1$ many vertex disjoint odd cycles of length $2r+1$.

On the other hand, Erdős, Hajnal, and Szemerédi proved that there is a $G$ with chromatic number $\aleph_1$ such that $h_G(n)\ll n^{3/2}$. In [Er81] Erdős conjectured that this can be improved to $\ll n^{1+\epsilon}$ for every $\epsilon>0$.

See also [74].

Conjectured by Erdős and Hajnal. In arrow notation, this is asking where $\alpha \to (\alpha,3)^2$ implies $\alpha \to (\alpha, n)^2$ for every finite $n$.

The answer is no, as independently shown by Schipperus [Sc99] (published in [Sc10]) and Darby [Da99].

For example, Larson [La00] has shown that this is false when $\alpha=\omega^{\omega^2}$ and $n=5$. There is more background and proof sketches in Chapter 2.9 of [HST10], by Hajnal and Larson.

A problem of Erdős from 1954. Sierpinski and Kurepa independently proved that this is true for $2$-colours. Erdős proved that this is true under the continuum hypothesis that $\mathfrak{c}=\aleph_1$, and offered \$100 for deciding what happens if the continuum hypothesis is not assumed.

Shelah proved that it is consistent that the answer is negative, although with a very large value of $\mathfrak{c}$. It remains open whether it is consistent to have a negative answer assuming $\mathfrak{c}=\aleph_2$.

Erdős and Hajnal [ErHa60] proved the existence of arbitrarily large finite independent sets (under the assumptions in the first problem).

Erdős writes in [Er61] that Gladysz has proved the existence of an independent set of size $2$ in the second question, but I cannot find a reference.

Hechler [He72] has shown the answer to the first question is no, assuming the continuum hypothesis.

A problem of Erdős and Rado. For comparison, Specker [Sp57] proved this property holds when $\alpha=\omega^2$ and false when $\alpha=\omega^n$ for $3\leq n<\omega$ (a question of Erdős for which he offered \$20).

This is true, and was proved by Chang [Ch72]. Milner modified Chang's proof to prove that this remains true if we replace $K_3$ by $K_m$ for all finite $m<\omega$ (a shorter proof was found by Larson [La73]).

See also [591] and [592].

For comparison, Specker [Sp57] proved this property holds when $\alpha=\omega^2$ and false when $\alpha=\omega^n$ for $3\leq n<\omega$. Chang proved this property holds when $\alpha=\omega^\omega$ (see [590]).

See [592] for the general case.

This property holds for $\beta=2$ and not for $3\leq \beta <\omega$ (Specker [Sp57]) and for $\beta=\omega$ (Chang [Ch72]).

The first open case is $\beta=\omega^2$ (see [591]). Galvin and Larson [GaLa74] have shown that if $\beta\geq 3$ has this property then $\beta$ must be 'additively indecomposable', so that in particular $\beta=\omega^\gamma$ for some $\gamma<\omega_1$. Galvin and Larson conjecture that every $\beta\geq 3$ of this form has this property.

See also [590].

Similar problems were investigated by Erdős, Galvin, and Hajnal [EGH75]. Erdős claims that for graphs the problem is completely solved: a graph of chromatic number $\geq \aleph_1$ must contain all finite bipartite graphs but need not contain any fixed odd cycle.

A problem of Erdős and Hajnal (who proved this for chromatic number $\geq \aleph_2$). This was proved by Erdős, Hajnal, and Shelah [EHS74].

See also [593] and [737].

A problem of Erdős and Hajnal. Folkman [Fo70] and Nešetřil and Rödl [NeRo75] have proved that for every $n\geq 1$ there is a graph $G$ which contains no $K_4$ and is not the union of $n$ triangle-free graphs.

See also [582] and [596].

Erdős and Hajnal originally conjectured that there are no such $G_1,G_2$, but in fact $G_1=C_4$ and $G_2=C_6$ is an example. Indeed, for this pair Nešetřil and Rödl established the first property and Erdős and Hajnal the second (in fact every $C_4$-free graph is a countable union of trees).

Whether this is true for $G_1=K_4$ and $G_2=K_3$ is the content of [595].

Erdős and Hajnal proved that $\omega_1^2 \to (\omega_1\omega,3)^2$. Erdős originally asked this with just the assumption that $G$ is $K_4$-free, but Baumgartner proved that $\omega_1^2 \not\to (\omega_1\omega, K_{\aleph_0,\aleph_0})^2$.

Sometimes known as the Erdős-Menger conjecture. When $G$ is finite this is equivalent to Menger's theorem. Erdős was interested in the case when $G$ is infinite.

This was proved by Aharoni and Berger [AhBe09].

A problem of Erdős, Hajnal, and Milner [EHM70], who proved this is true for $\alpha < \omega_1^{\omega+2}$.

In [Er82e] Erdős offers \$250 for showing what happens when $\alpha=\omega_1^{\omega+2}$ and \$500 for settling the general case.

Larson [La90] proved this is true for all $\alpha<2^{\aleph_0}$ assuming Martin's axiom.

A problem of Komjáth. The existence of such a $2$-colouring is sometimes known as Property B.

A problem of Komjáth. If instead we have $\lvert A_i\cap A_j\rvert \neq 1$ then Komjáth showed that this is possible with at most $\aleph_0$ colours.

A problem of Erdős and Hajnal [ErHa58], who proved that if $\lvert X\rvert <\aleph_\omega$ then the answer is no. Erdős suggests in [Er99] that this problem is 'perhaps undecidable'.