2. Proof of Theorem 1.3

In this section we prove Theorem 1.3, assuming the following lemma. Given $k \in{\mathbb{Z}}$ , we write $\log ^k D \,:\!=\, (\log D)^k$ , where the logarithm is base $e$ . Given $\Lambda, D, C, p \in{\mathbb{R}}$ , we also define

\begin{align*}{\Lambda _{\mathbb{E}}}(\Lambda, D, C, p) &\,:\!=\, \left (1 - \frac{p}{\Lambda }\right )^{DC}\Lambda && \mathrm{and}\\{D_{{\mathbb{E}}}}(\Lambda, D, C, p) &\,:\!=\, \left ((1 - p)\left (1 - \frac{p}{\Lambda }\right )^{D(C-1)} + p\left (1 - \left (1 - \frac{p}{\Lambda }\right )^{DC}\right )\right ) D. \end{align*}

We prove Lemma 2.1 in Section 4 by analysing a random colouring procedure, but we already explain some of the ideas involved in the proof now. In our random colouring procedure, we randomly assign each vertex $v \in V(G)$ a colour $\psi (v) \in L(v)$ uniformly at random from its list, and we ‘activate’ vertices independently at random with probability $p$ . (We really only need to assign colours to activated vertices, but for technical reasons it is convenient to define the procedure this way.) For each vertex, we remove any colour from its list assigned to an activated neighbour, and we let $X$ be the set of activated vertices whose colour was not assigned to any of its activated neighbours. We obtain $L^{\prime}$ by further truncating each vertex’s list to have the desired size, and we show that with nonzero probability, $X$ , $L^{\prime}$ , and $\phi \,:\!=\, \psi |_X$ satisfy (2.1.1) and (2.1.2).

To that end, assuming without loss of generality that $d_{G_i, L}(v, c) = D$ for every $i \in [m]$ and $v \in V(G_i)$ with $c \in L(v)$ (see Proposition 4.1), it is straightforward to show that a vertex ‘keeps’ a given colour in its list with probability $(1 - p/ \Lambda )^{DC}$ (see Proposition 4.2). Thus, after applying this procedure, ${\Lambda _{\mathbb{E}}}(\Lambda, D, C, p)$ is the expected number of remaining available colours for each vertex, and ${D_{{\mathbb{E}}}}(\Lambda, D, C, p)$ is an upper bound on the expected colour degree of each pair of vertex and colour from its list (see (4) and Lemma 4.3). We show that with very high probability (at least $1 - \exp \left (-D^{1/4}\right )$ – see Lemma 4.5), the number of remaining colours available for a given vertex, and the colour degree in $G - X$ of a given pair of vertex and colour, are not significantly larger than ${\Lambda _{\mathbb{E}}}(\Lambda, D, C, p)$ and ${D_{{\mathbb{E}}}}(\Lambda, D, C, p)$ , respectively. We complete the proof by applying the Lovász Local Lemma.

To prove Theorem 1.3, we iteratively apply Lemma 2.1 $O(C{\varepsilon }^{-1}\log D)$ times, each time reapplying the lemma with $p = \log ^{-1}D$ and with $G_1 - X, \dots, G_m - X$ and $L^{\prime}$ playing the roles of $G_1, \dots, G_m$ and $L$ , respectively, before completing the colouring with Lemma 2.3 below. The requirement $L^{\prime}(v) \subseteq L(v) \setminus \{\phi (u) \,:\, u \in N_G(v) \cap X\}$ ensures that adjacent vertices are not assigned the same colour in different iterations, so each iteration extends a proper partial colouring of $G$ .

This approach – known as the ‘nibble method’ or the ‘semi-random method’ – has led to numerous important developments in graph colouring (see [Reference Kang16, Reference Molloy and Reed21]). Our random colouring procedure was in fact already used by Reed and Sudakov [Reference Reed and Sudakov25] to prove the special case of Theorem 1.3 when $C = 1$ ; however, our analysis of the procedure is different even in this special case, drawing ideas from a recent proof of Kang and Kelly [Reference Kang and Kelly18], and we need new ideas to analyse the procedure for $C \gt 1$ . Recall that Theorem 1.3 also implies Kahn’s [Reference Kahn14] bound on the chromatic index of linear uniform hypergraphs and that Theorem 5.2 implies Molloy and Postle’s [Reference Molloy and Postle19] generalization of Kahn’s result to correspondence colouring. These edge-colouring results are also proved with a semi-random approach, but the random colouring procedures used in these proofs are slightly different from ours. Since vertices may lose colours unnecessarily in our procedure – it would suffice to only remove colours from a vertex’s list that were assigned to a neighbour in $X$ , but this version of the procedure would be more challenging to analyse – Reed and Sudakov [Reference Reed and Sudakov25] called the procedure ‘wasteful’ (see also [Reference Molloy and Reed21, Chapter 12]). Nevertheless, this wastefulness is negligible because our activation probability $p$ is small. However, Kahn [Reference Kahn14] did not consider activation probabilities and consequently could not afford to use a wasteful variant of his procedure. Molloy and Postle’s [Reference Molloy and Postle19] proof does consider activation probabilities but is still different from ours, even in the special case of edge-colouring; in particular, vertices (of the line graph) may be assigned multiple colours in their colouring procedure.

Let us now explain how we use Lemma 2.1 to prove Theorem 1.3. Crucially, (2.1.1) and (2.1.2) together imply that after each iteration of Lemma 2.1, the ratio of the number of remaining available colours for each vertex to the maximum remaining colour degree in each $G_i$ , while initially only $1 +{\varepsilon }$ , improves by a factor of at least $1 +{\varepsilon } p/ 4$ . Moreover, the number of available colours does not decrease too much in each iteration. The following proposition makes this calculation precise.

Proposition 2.2. For every $C\geq 1$ and $0 \lt{\varepsilon } \lt 1$ there exists $D_{2.2}$ such that the following holds for every $D \geq D_{2.2}$ . If $\log ^{-1}D \geq p \geq \log ^{-2}D$ and $10CD \geq \Lambda \geq (1 +{\varepsilon })D$ , then

(2) \begin{equation} \frac{{\Lambda _{\mathbb{E}}}(\Lambda, D, C, p) - \Lambda ^{4/5}}{{D_{{\mathbb{E}}}}(\Lambda, D, C, p) + D^{4/5}} \geq (1 +{\varepsilon } p/ 4)\frac{\Lambda }{D} \end{equation}

and

(3) \begin{equation} {D_{{\mathbb{E}}}}(\Lambda, D, C, p) \geq (1 - pC)D. \end{equation}

Proof. First we prove (3). Clearly,

\begin{equation*} (1 - p)\left (1 - \frac {p}{\Lambda }\right )^{D(C-1)}D \geq (1 - p)(1 - p(C - 1))D \geq (1 - pC)D, \end{equation*}

and (3) follows immediately, since $p\left (1 - \left (1 - p/ \Lambda \right )^{DC}\right ) \gt 0$ .

Now we prove (2). By (3), since $\Lambda \leq 10CD$ and $\log ^{-1} D \geq p \geq \log ^{-2}D$ ,

\begin{equation*} \frac {\Lambda ^{4/5}}{{D_{{\mathbb {E}}}}(\Lambda, D, C, p) + D^{4/5}} \leq \frac {10CD^{4/5}}{(1 - pC)D} \leq \frac {{\varepsilon } p}{100}\cdot \frac {\Lambda }{D}. \end{equation*}

Let

\begin{equation*} W \,:\!=\, (1 - p)\left (1 - \frac {p}{\Lambda }\right )^{-D} + p\left (\left (1 - \frac {p}{\Lambda }\right )^{-DC} - 1\right ) + \left (1 - \frac {p}{\Lambda }\right )^{-DC}D^{-1/5}, \end{equation*}

and note that

\begin{equation*} \frac {{\Lambda _{\mathbb {E}}}(\Lambda, D, C, p)}{{D_{{\mathbb {E}}}}(\Lambda, D, C, p) + D^{4/5}} = \frac {1}{W}\cdot \frac {\Lambda }{D}. \end{equation*}

Since $1 + x \leq (1 + x/ n)^n$ for $x \leq |n|$ and $\Lambda \geq (1 +{\varepsilon })D$ , we have

\begin{align*} W & \leq \frac {1 - p}{1 - p/(1 + {\varepsilon })} + p\left (\frac {1}{1 - pC/(1 + {\varepsilon })} - 1\right ) + \frac {D^{-1/5}}{1 - pC/(1 + {\varepsilon })} \leq 1 - p + \frac {p}{1 + {\varepsilon }} +O(p^2)\\& \leq 1 - \frac {p{\varepsilon }}{3}. \end{align*}

Since $(1 - p{\varepsilon }/ 3)^{-1} \geq 1 + p{\varepsilon }/ 3$ , by combining the inequalities above, we have that the left side of (2) is at least

\begin{equation*} \left (1 + p{\varepsilon }/ 3 - {\varepsilon } p/100\right )\frac {\Lambda }{D} \geq (1 + {\varepsilon } p/ 4)\frac {\Lambda }{D}, \end{equation*}

as desired.

We apply Lemma 2.1 iteratively until the ratio of the number of remaining available colours to the maximum remaining colour degree reaches $8CD$ : (2) will imply that this ratio improves by a factor of at least $1 +{\varepsilon }/ (4\log D)$ , so that we only need at most $33C{\varepsilon }^{-1} \log D$ iterations, and (3) will ensure that carrying out this many iterations is indeed possible. After this process, we can finish with the following result of Reed [Reference Reed24] (which is proved via a simple application of the Lovász Local Lemma).

Now we prove Theorem 1.3.

Proof of Theorem 1.3 . Without loss of generality, we may assume that ${\varepsilon } \lt 1$ . Let $p \,:\!=\, \log ^{-1} D$ , $D_0 \,:\!=\, D$ and $\Lambda _0 \,:\!=\, (1 +{\varepsilon })D_0$ , and for each integer $0\le i \le 33C/({\varepsilon } p)$ , let

\begin{align*} \Lambda _{i + 1} &\,:\!=\,{\Lambda _{\mathbb{E}}}(\Lambda _i, D_i, C, p) - \Lambda _i^{4/5} && \mathrm{and}\\ D_{i + 1} &\,:\!=\,{D_{{\mathbb{E}}}}(\Lambda _i, D_i, C, p) + D_i^{4/5}. \end{align*}

Applying Proposition 2.2 inductively, for every integer $0\le i \le 33C/({\varepsilon } p)$ , by (3) we have

\begin{equation*} D_{i} \ge \left (1 - pC\right )^{33C/({\varepsilon } p)}D_0 \geq e^{-33 C^2/ {\varepsilon }}(1 - 33C^3p/ {\varepsilon })D_0 \geq e^{-34 C^2/ {\varepsilon }}D_0, \end{equation*}

where the second inequality uses that $(1 + x/ n)^n \geq e^x(1 - x^2/ n)$ for $|x| \leq n$ . By (2),

\begin{equation*} \frac {\Lambda _{i}}{D_{i}} \geq \left (1 + \frac {{\varepsilon } p}{4}\right )\frac {\Lambda _{i-1}}{D_{i-1}}.\end{equation*}

Moreover, we have

\begin{equation*} \frac {\lceil \Lambda _{\lfloor 33C/({\varepsilon } p)\rfloor } \rceil }{D_{\lfloor 33C/({\varepsilon } p)\rfloor }} \geq \left (1 + \frac {{\varepsilon } p}{4}\right )^{\lfloor 33C/({\varepsilon } p) \rfloor }\frac {\Lambda _0}{D_0} \geq 8C. \end{equation*}

In particular, there exists an integer $0 \lt i^* \leq 33C/ ({\varepsilon } p)$ such that $\lceil \Lambda _{i^*}\rceil/ D_{i^*} \geq 8C$ and $\lceil \Lambda _{i^* - 1}\rceil/ D_{i^* - 1} \lt 8C$ . We may assume $D$ is sufficiently large so that $D_{i^*} \geq D_{2.1}$ .

By (iv), we may assume without loss of generality that $|L(v)| = \lceil \Lambda _0\rceil$ for every $v \in V(G)$ , since we can truncate each list until equality holds. Now let $G_{j,0} \,:\!=\, G_j$ for each $j \in [m]$ , let $H_0 \,:\!=\, G$ , and let $L_0 \,:\!=\, L$ . Due to the above calculations, inductively by Lemma 2.1, for each integer $0\le i \lt i^*$ , there is a set $X_i \subseteq V(H_i)$ and an $L_i|_{X_i}$ -colouring $\phi _i$ of $H_i[X_i]$ and a list assignment $L_{i+1}$ for $H_{i+1}\,:\!=\, H_i - X_i$ satisfying $L_{i+1}(v) \subseteq L_i(v) \setminus \{\phi _i(u) \,:\, u \in N_{H_i}(v) \cap X_i\}$ such that

• $\Delta \left(G_{j, i + 1}, L_{i + 1}|_{V(G_{j, i+1})}\right) \leq D_{i + 1}$ for each $j \in [m]$ and

• $|L_{i + 1}(v)| = \lceil \Lambda _{i + 1}\rceil \ge (1+{\varepsilon }) D_{i+1}$ ,

where $G_{j, i + 1} \,:\!=\, G_{j, i} - X_i$ .

Let $X \,:\!=\, \bigcup _{i=1}^{i^*-1} X_i$ , let $\phi (v) \,:\!=\, \phi _i(v)$ if $v \in X_i$ for some integer $0\le i \lt i^*$ , let $G^{\prime} \,:\!=\, H_{i^*}$ , and let $L^{\prime} \,:\!=\, L_{i^*}$ . By construction, $\phi$ is an $L|_X$ -colouring of $G[X]$ , and by the choice of $i^*$ , we have

• $|L^{\prime}(v)| \geq 8C D_{i^*}$ and

• $\Delta (G^{\prime}, L^{\prime}) \leq C\max \{\Delta \left(G_{j, i^*}, L_{i^*}|_{V\left(G_{j,i^*}\right)}\right) :{j \in [m]}\} \leq CD_{i^*}$ .

Therefore by Lemma 2.3, $G^{\prime}$ has an $L^{\prime}$ -colouring $\phi^{\prime}$ , and we can combine $\phi$ and $\phi^{\prime}$ to obtain an $L$ -colouring of $G$ , as desired.

3. Probabilistic tools

In this section we provide some probabilistic tools used in the proof of Lemma 2.1 in Section 4. The first such tool is the Lovász Local Lemma.

Next we need a concentration inequality of Bruhn and Joos [Reference Bruhn and Joos5], derived from Talagrand’s inequality [Reference Talagrand26]. To that end, we introduce the following definition.

Theorem 3.3 (Bruhn and Joos [Reference Bruhn and Joos5]). Let $(( \boldsymbol{\Omega} _i, \boldsymbol{\Sigma} _i,{\mathbb{P}}_i))$ be probability spaces, let $( \boldsymbol{\Omega}, \boldsymbol{\Sigma},{\mathbb{P}})$ be their product space, and let $ \boldsymbol{\Omega} ^*\in \boldsymbol{\Sigma}$ be a set of exceptional outcomes. Let ${\textbf X} \,:\, \boldsymbol{\Omega} \rightarrow{\mathbb{R}}$ be a non-negative random variable, let $M \,:\!=\, \max \{\sup{\textbf X}, 1\}$ , and let $ \delta \geq 1$ . If ${{\mathbb{P}}\left [ \boldsymbol{\Omega} ^*\right ] } \leq M^{-2}$ and $\textbf X$ has upward $(s, \delta )$ -certificates, then for $t \gt 50 \delta \sqrt s$ ,

\begin{equation*} {{\mathbb {P}}\left [|{\textbf X} - {{\mathbb {E}}\left [{\textbf X}\right ] }| \geq t\right ] } \leq 4\exp \left (-\frac {t^2}{16 \delta ^2s}\right ) + 4{{\mathbb {P}}\left [ \boldsymbol{\Omega} ^*\right ] }. \end{equation*}

4. Proof of Lemma 2.1

This section is devoted to the proof of Lemma 2.1. It will be convenient to assume that equality holds in (ii) and (iii) of Theorem 1.3 and that moreover $d_{G_i, L}(v, c) = D$ for every $v \in V(G)$ and $c \in L(v)$ , so we first prove the following proposition which enables us to consider an embedding of $G$ for which these properties hold.

The proof of Proposition 4.1 can be found in the appendix. For the remainder of this section, we let $C,{\varepsilon } \gt 0$ , and we let $D$ be sufficiently large, we let $p$ satisfy $\log ^{-1} D \geq p \geq \log ^{-2}D$ , and we let $G\,:\!=\, \bigcup _{i=1}^mG_i$ with list assignment $L$ satisfying the hypotheses of Lemma 2.1. For convenience, we assume $\Lambda$ and $D$ are integers where it does not affect the argument. By Proposition 4.1, we may assume without loss of generality that $|\{i \in [m] \,:\, v \in V(G_i)\}| = C$ for every $v\in V(G)$ and that $d_{G_i, L}(v, c) = D$ for every $i \in [m]$ and $v \in V(G)$ with $c \in L(v)$ .

For every pair $(A, \psi )$ satisfying $A \subseteq V(G)$ and $\psi (v) \in L(v)$ for each $v\in V(G)$ ,

• let $L_{A, \psi }(v) \,:\!=\, L(v) \setminus \{\psi (u) \,:\, u \in N_G(v)\cap A\}$ for every $v \in V(G)$ , and

• let $X_{A, \psi } \,:\!=\, \{v \in A \,:\, \psi (v) \in L_{A, \psi }(v)\}$ .

Note that $X_{A, \psi } = A \setminus \bigcup _{v \in A}\{u \in N_G(v) \,:\, \psi (u) = \psi (v)\}$ .

We will consider the probability space $( \boldsymbol{\Omega}, \boldsymbol{\Sigma},{\mathbb{P}})$ of such pairs $(A, \psi )$ where

• each vertex in $G$ is in $A$ independently and uniformly at random with probability $p$ and

• $\psi (v) \in L(v)$ is chosen independently and uniformly at random for each $v \in V(G)$ .

Note that $\left( \boldsymbol{\Omega}, \boldsymbol{\Sigma},{\mathbb{P}}\right)$ is the product space of $\left( \boldsymbol{\Omega} ^{\mathrm{act}}_{v}, \boldsymbol{\Sigma} ^{\mathrm{act}}_{v},{\mathbb{P}}^{\mathrm{act}}_{v}\right)$ and $\left( \boldsymbol{\Omega} ^{\mathrm{col}}_{v}, \boldsymbol{\Sigma} ^{\mathrm{col}}_{v},{\mathbb{P}}^{\mathrm{col}}_{v}\right)$ taken over all $v \in V(G)$ , where $\left( \boldsymbol{\Omega} ^{\mathrm{act}}_{v}, \boldsymbol{\Sigma} ^{\mathrm{act}}_{v},{\mathbb{P}}^{\mathrm{act}}_{v}\right)$ models whether $v \in A$ and $\left( \boldsymbol{\Omega} ^{\mathrm{col}}_{v}, \boldsymbol{\Sigma} ^{\mathrm{col}}_{v},{\mathbb{P}}^{\mathrm{col}}_{v}\right)$ models the choice of $\psi (v)$ . We will use this fact to apply Theorem 3.3.

We will use the Lovász Local Lemma to show that with nonzero probability, $X = X_{A, \psi }$ and $L^{\prime} = L_{A, \psi }|_{V(G)\setminus X}$ (with lists possibly truncated) satisfy the lemma with $\phi = \psi |_X$ . To that end, we define the following random variables for each vertex $v\in V(G)$ and $c\in L(v)$ and $i \in [m]$ :

• $ \boldsymbol{\Lambda}_v(A, \psi ) \,:\!=\, \left |L_{A,\psi }(v)\right |$ ,

• $ \textbf{D}_{v, c, i}(A, \psi ) \,:\!=\, \left |\left \{u \in N_{G_i}(v) \,:\, c \in L_{A, \psi }(u)\right \} \setminus X_{A, \psi }\right |$ ,

• $ \textbf{Y}_{v, c, i}(A, \psi ) \,:\!=\, \left |\left \{u \in N_{G_i}(v) \cap A \,:\, c \in L(u)\right \} \setminus X_{A, \psi }\right |$ ,

• $ \textbf{R}_{v, c, i}(A, \psi ) \,:\!=\, \left |\left \{u \in N_{G_i}(v) \setminus A \,:\, c \in L(u)\right \} \setminus \bigcup _{u \in A\setminus V(G_i) \,:\, \psi (u) = c}N_G(u)\right |$ .

Note that

(4) \begin{align} \textbf{D}_{v, c, i}(A, \psi ) & = \textbf{Y}_{v, c, i}(A, \psi ) - |\{u \in (N_{G_i}(v) \cap A)\setminus X_{A, \psi } \,:\, c \in L(u)\} \setminus \bigcup _{u \in A \,:\, \psi (u) = c}N_G(u)\}| \nonumber\\& \quad + |\{u \in N_{G_i}(v) \setminus A \,:\, c \in L(u)\} \setminus \bigcup _{u \in A \,:\, \psi (u) = c}N_G(u)\}| \leq \textbf{Y}_{v, c, i}(A, \psi ) + \textbf{R}_{v, c, i}(A, \psi ). \end{align}

To apply the Lovász Local Lemma, we first show that for each $v \in V(G)$ , we have that $ \boldsymbol{\Lambda}_v$ is at least the quantity in (2.1.1) with high probability, and for each $c \in L(v)$ and $i \in [m]$ such that $v \in V(G_i)$ , we have that $ \textbf{D}_{v, c, i}$ is at most the quantity in (2.1.2) with high probability. To that end, we first compute the expectation of these random variables and then apply Theorem 3.3. However, it is not possible to apply Theorem 3.3 to $ \textbf{D}_{v, c, i}$ directly to obtain any meaningful bound. (Consider the case $C = m = 1$ when $G_1$ is complete. Either no vertex in $A$ is assigned $c$ , in which case $c \in L_{A, \psi }(u)$ for every vertex $u$ , or some vertex in $A$ is assigned $c$ , in which case $c \notin L_{A, \psi }(u)$ for all $u \in V(G)$ .) Nevertheless, we can apply Theorem 3.3 to both $ \textbf{Y}_{v, c, i}$ and $ \textbf{R}_{v, c, i}$ and moreover show that $ \textbf{Y}_{v, c, i} + \textbf{R}_{v, c, i}$ is at most the quantity in (2.1.2) with high probability. Hence, (4) implies the required bound for $ \textbf{D}_{v, c, i}$ as well.

In order to compute the expected values of $ \boldsymbol{\Lambda}_v$ , $ \textbf{Y}_{v, c, i}$ , and $ \textbf{R}_{v, c, i}$ , we need the following simple proposition.

Proposition 4.2. For every $i \in [m]$ , $v\in V(G_i)$ , and $c \in L(v)$ ,

(5) \begin{equation} {{\mathbb{P}}\left [\not \exists u \in N_{G_i}(v) \cap A \text{ such that }\psi (u) = c\right ] } = \left (1 - \frac{p}{\Lambda }\right )^D. \end{equation}

Moreover, for every $v\in V(G)$ and $c\in L(v)$ ,

(6) \begin{equation} {{\mathbb{P}}\left [c \in L_{A, \psi }(v)\right ] } = \left (1 - \frac{p}{\Lambda }\right )^{DC}. \end{equation}

Proof. First we prove (5). For each $u \in N_{G_i}(v)$ with $c \in L(u)$ , we have ${{\mathbb{P}}\left [u \in A\right ] } = p$ and ${{\mathbb{P}}\left [\psi (u) = c\right ] } = 1/ \Lambda$ , and the events that $u \in A$ and that $\psi (u) = c$ are independent. Moreover, these events are mutually independent for all such $u$ , so

\begin{equation*} {{\mathbb {P}}\left [\not \exists u \in N_{G_i}(v) \cap A \text { s.t. }\psi (u) = c\right ] } = \prod _{u \in N_{G_i}(v) \,:\, c \in L(u)}\left (1 - {{\mathbb {P}}\left [u \in A\right ] }\cdot {{\mathbb {P}}\left [\psi (u) = c\right ] }\right ) = \left (1 - \frac {p}{\Lambda }\right )^D, \end{equation*}

as desired.

Now we prove (6). For every $v\in V(G)$ and $c \in L(v)$ , we have $c \in L_{A, \psi }(v)$ if and only if there is no $u \in N_{G_i}(v) \cap A$ with $\psi (u) = c$ for every $i$ for which $v \in V(G_i)$ . By (i), these events are mutually independent for all such $i$ (of which there are $C$ by (ii)), so by (5),

\begin{equation*} {{\mathbb {P}}\left [c\in L_{A, \psi }(v)\right ] } = \prod _{i \in [m] \,:\, v \in V(G_i)}{{\mathbb {P}}\left [\not \exists u \in N_{G_i}(v) \cap A \text { s.t. }\psi (u) = c\right ] } = \left (1 - \frac {p}{\Lambda }\right )^{DC}, \end{equation*}

as desired.

Now we compute the expectations of our random variables.

Lemma 4.3. Every vertex $v\in V(G)$ satisfies

(7) \begin{equation} {{\mathbb{E}}\left [ \boldsymbol{\Lambda}_v\right ] } = \left (1 - \frac{p}{\Lambda }\right )^{DC}\Lambda ={\Lambda _{\mathbb{E}}}(\Lambda, D, C, p). \end{equation}

Moreover, for every $i \in [m]$ , $v \in V(G_i)$ , and $c \in L(v)$ ,

(8) \begin{align}{{\mathbb{E}}\left [ \textbf{Y}_{v, c, i}\right ]} = p\left (1 - \left (1 - \frac{p}{\Lambda }\right )^{DC}\right )D\,\text{and} \end{align}

(9) \begin{align}{{\mathbb{E}}\left [ \textbf{R}_{v, c, i}\right ]} = (1 - p)\left (1 - \frac{p}{\Lambda }\right )^{D(C - 1)}D. \end{align}

Proof. By the linearity of expectation, we have ${{\mathbb{E}}\left [ \boldsymbol{\Lambda}_v\right ] } = \sum _{c\in L(v)}{{\mathbb{P}}\left [c \in L_{A, \psi }(v)\right ] }$ , so (7) follows from (6).

Again by the linearity of expectation, we have

\begin{equation*} {{\mathbb {E}}\left [ \textbf {Y}_{v, c, i}\right ] } = \sum _{u \in N_{G_i}(v) \,:\, c \in L(u)}{{\mathbb {P}}\left [u \in A \text { and } u \notin X_{A, \psi }\right ] }. \end{equation*}

Note that ${{\mathbb{P}}\left [u \in A \text{ and } u \notin X_{A, \psi }\right ] } ={{\mathbb{P}}\left [u \in A \text{ and } \psi (u) \notin L_{A, \psi }(u)\right ] }$ , and the events that $u \in A$ and that $\psi (u) \notin L_{A, \psi }(u)$ are independent. We have ${{\mathbb{P}}\left [u \in A\right ] } = p$ and ${{\mathbb{P}}\left [\psi (u) \notin L_{A, \psi }(u)\right ] } = 1 - \left (1 - p/ \Lambda \right )^{DC}$ by (6), so (8) follows from the equation above.

By the linearity of expectation, we have

\begin{equation*} {{\mathbb {E}}\left [ \textbf {R}_{v, c, i}\right ] } = \sum _{u \in N_{G_i}(v) \,:\, c \in L(u)}{{\mathbb {P}}\left [u \notin A \text { and } \psi (w) \neq c \forall w \in N_{G_j}(u) \cap A, j\in [m]\setminus \{i\}\right ] }. \end{equation*}

By (5), ${{\mathbb{P}}\left [\psi (w) \neq c \forall w \in N_{G_j}(u) \cap A, j\in [m]\setminus \{i\}\right ] } = \left (1 - p/ \Lambda \right )^{D(C - 1)}$ . Since the events that $u \notin A$ and $\psi (w) \neq c \forall w \in N_{G_j}(u)\cap A, j\in [m]\setminus \{i\}$ are independent, (9) follows from the equation above.

Now we need to show that the random variables in Lemma 4.3 are close to their expectation with high probability. We use Theorem 3.3, the ‘exceptional outcomes’ version of Talagrand’s Inequality. To that end, we define an exceptional outcome for each vertex and show that it is unlikely. First, for each vertex $v\in V(G)$ and $c\in L(v)$ and $i \in [m]$ , we define the random variable

• $ \textbf{F}_{v, c, i}(A, \psi ) \,:\!=\, \left |\left \{u \in N_{G_i}(v) \,:\, \psi (u) = c\right \}\right |$ .

Then, for each vertex $v\in V(G)$ and $i \in [m]$ , we define

• $ \boldsymbol{\Omega} ^*_{v, i} = \{(A, \psi ) \,:\, \exists u \in \{v\} \cup N_{G_i}(v) \cup N^2_{G_i}(v), \exists c\in L(u), \textbf{F}_{u, c, i}(A, \psi ) \geq \log D \}$ ,

where $N^2_{G_i}(v)$ is the set of vertices at distance two from $v$ in $G_i$ . To apply Theorem 3.3 to $ \textbf{Y}_{v, c, i}$ , we will show that $ \textbf{Y}_{v, c, i}$ has upward $(s, \delta )$ -certificates with respect to exceptional outcomes $ \boldsymbol{\Omega} ^*_{v, i}$ with $s = 4D$ and $\delta = \log D$ . It turns out that we need to consider these exceptional outcomes because, for example, if there is a set of more than $\log D$ vertices in $N_{G_i}(v) \cap A$ that are all assigned the same colour, then changing the outcome of a single trial (in particular the trial determining either $\psi (u)$ or whether $u \in A$ for some $u \in \{v\} \cup N_{G_i}(v) \cup N^2_{G_i}(v)$ for which $ \textbf{F}_{u, c, i}(A, \psi ) \geq \log D$ for some $c\in L(u)$ ) can affect the value of $ \textbf{Y}_{v, c, i}$ by more than $\log D$ (see the final part of the proof of Lemma 4.5 for more details). We remark that the event $ \boldsymbol{\Omega} ^*_{v, i}$ contains more outcomes than is necessary; it is simply more convenient to consider $ \boldsymbol{\Omega} ^*_{v, i}$ , and the argument adapts more easily for correspondence colouring in Section 5.

Now we bound the probability of these exceptional events. For this, we assume without loss of generality that $uv \in E(G)$ only if $L(u) \cap L(v) \neq \emptyset$ , so each $G_i$ has maximum degree at most $\Lambda D$ .

Proposition 4.4. Every vertex $v\in V(G)$ and $i \in [m]$ satisfies

\begin{equation*} {{\mathbb {P}}\left [ \boldsymbol{\Omega} ^*_{v, i}\right ] } \leq 11^3 C^3D^5\left (\log D\right )^{- \log D}. \end{equation*}

Proof. First we bound the probability that $ \textbf{F}_{u, c, i}$ is too large for each $u \in V(G)$ and $c\in L(u)$ , as follows:

\begin{equation*} {{\mathbb {P}}\left [ \textbf {F}_{u, c, i} \geq \log D\right ] } \leq \sum _{i = \lceil \log D\rceil }^{D}\binom {D}{i}\left (\frac {1}{\Lambda }\right )^i \leq \sum _{i=\lceil \log D\rceil }^{D}\left (\frac {eD}{i\Lambda }\right )^i \leq \sum _{i=\lceil \log D\rceil }^{D}\left (\frac {e}{i}\right )^i. \end{equation*}

Since each term in the sum is at most $(e/\log D)^{\log D}$ and there are at most $D$ terms, it follows that

\begin{equation*} {{\mathbb {P}}\left [ \textbf {F}_{u, c, i} \geq \log D\right ] } \leq D^2\left (\log D\right )^{- \log D}. \end{equation*}

Thus, combining the above inequality with the Union Bound, we have ${{\mathbb{P}}\left [ \boldsymbol{\Omega} ^*_{v,i}\right ] } \leq \Lambda (1 + \Lambda D + \Lambda ^2D^2)\left (\log D\right )^{- \log D} \leq 11^3 C^3D^5\left (\log D\right )^{- \log D}$ , as desired.

Combining Theorem 3.3 and Proposition 4.4, we show that $ \boldsymbol{\Lambda}_v$ , $ \textbf{Y}_{v, c, i}$ , and $ \textbf{R}_{v, c, i}$ are close to their expectation with high probability in the following lemma.

Lemma 4.5. Every vertex $v\in V(G)$ satisfies

(10) \begin{equation} {{\mathbb{P}}\left [\left | \boldsymbol{\Lambda}_v -{{\mathbb{E}}\left [ \boldsymbol{\Lambda}_v\right ] }\right | \gt D^{2/3}\right ] } \leq \exp \left (-D^{1/4}\right ). \end{equation}

Moreover, if $v \in V(G_i)$ and $c \in L(v)$ , then

(11) \begin{equation}{{\mathbb{P}}\left [\left | \textbf{Y}_{v, c, i} -{{\mathbb{E}}\left [ \textbf{Y}_{v, c, i}\right ]}\right | \gt D^{2/3}\right ]} \quad \leq \exp \left (-D^{1/4}\right )\qquad \text{and} \end{equation}

(12) \begin{equation}{{\mathbb{P}}\left [\left | \textbf{R}_{v, c, i} -{{\mathbb{E}}\left [ \textbf{R}_{v, c, i}\right ] }\right | \gt D^{2/3}\right ]} \quad \leq \exp \left (-D^{1/4}\right ). \qquad\quad \end{equation}

Proof. First we prove (10). We apply Theorem 3.3 with exceptional outcomes $ \boldsymbol{\Omega} ^* = \emptyset$ . To that end, we show that $\Lambda - \boldsymbol{\Lambda}_v$ has upward $(s, \delta )$ -certificates with respect to $ \boldsymbol{\Omega} ^*$ , where $s = 2\Lambda$ and $\delta = 1$ . Let $(A, \psi ) \in \boldsymbol{\Omega}$ . For every $c \in L(v) \setminus L_{A, \psi }(v)$ , there is at least one neighbour $u \in N_{G}(v)$ of $v$ such that $u \in A$ and $\psi (u) = c$ . Choose one such neighbour, and denote it by $u_c$ . Let $I_{A, \psi }$ index the trials determining whether $u_c \in A$ and $\psi (u_c) = c$ for each $c \in L(v)\setminus L_{A, \psi }(v)$ . Now if $(A^{\prime}, \psi^{\prime}) \in \boldsymbol{\Omega}$ differs from $(A, \psi )$ in at most $t$ of the trials indexed by $I_{A, \psi }$ (and differs arbitrarily for trials not indexed by $I_{A, \psi }$ ), then all but at most $t$ colours in $L(v) \setminus L_{A, \psi }(v)$ are also in $L(v) \setminus L_{A^{\prime}, \psi^{\prime}}(v)$ . Hence, $\Lambda - \boldsymbol{\Lambda}_v(A^{\prime}, \psi^{\prime}) \geq \Lambda - \boldsymbol{\Lambda}_v(A, \psi ) - t$ , and since $I_{A, \psi } \leq 2\Lambda = s$ , it follows that $\Lambda - \boldsymbol{\Lambda}_v$ has upward $(s, \delta )$ -certificates, as desired. Therefore by Theorem 3.3 with $t = D^{2/3}$ ,

\begin{equation*} {{\mathbb {P}}\left [\left | \boldsymbol{\Lambda}_v - {{\mathbb {E}}\left [ \boldsymbol{\Lambda}_v\right ] }\right | \gt D^{2/3}\right ] } \leq 4\exp \left (-\frac {D^{4/3}}{32\Lambda }\right ) \leq \exp \left (-D^{1/4}\right ), \end{equation*}

as desired.

Now we prove (12). We apply Theorem 3.3 with exceptional outcomes $ \boldsymbol{\Omega} ^* = \emptyset$ . To that end, we show that $D - \textbf{R}_{v, c, i}$ has upward $(s, \delta )$ -certificates with respect to $ \boldsymbol{\Omega} ^*$ , where $s = 3D$ and $\delta = 1$ . Let $(A, \psi ) \in \boldsymbol{\Omega}$ , and note that

\begin{equation*} D - \textbf {R}_{v, c, i}(A, \psi ) = |\{x \in N_{G_i}(v) \,:\, c \in L(x)\} \cap (A \cup \bigcup _{w \in A\setminus V(G_i) \,:\, \psi (w) = c}N_G(w))|. \end{equation*}

For every $u \in \{x \in N_{G_i}(v) \,:\, c \in L(x)\} \cap \bigcup _{w \in A\setminus V(G_i) \,:\, \psi (w) = c}N_G(w)$ , there is at least one neighbour $w \in N_{G_j}(u)$ of $u$ where $j \in [m]\setminus \{i\}$ such that $w \in A$ and $\psi (w) = c$ . Choose one such neighbour, and denote it by $w_u$ . By (i), the vertices $w_u$ are distinct for different choices of $u$ . Let $I_{A, \psi }$ index the trials determining whether $u \in A$ for each $u \in \{x \in N_{G_i}(v) \,:\, c \in L(x)\}$ and whether $w_u \in A$ and $\psi (w_u) = c$ for each $u \in \{x \in N_{G_i}(v) \,:\, c \in L(x)\} \cap \bigcup _{w \in A\setminus V(G_i) \,:\, \psi (w) = c}N_G(w)$ . Now if $(A^{\prime}, \psi^{\prime}) \in \boldsymbol{\Omega}$ differs from $(A, \psi )$ in at most $t$ of the trials indexed by $I_{A, \psi }$ (and differs arbitrarily for trials not indexed by $I_{A, \psi }$ ), all but at most $t$ vertices in $\{x \in N_{G_i}(v) \,:\, c \in L(x)\} \cap (A \cup \bigcup _{w \in A\setminus V(G_i) \,:\, \psi (w) = c}N_G(w))$ are in $\{x \in N_{G_i}(v) \,:\, c \in L(x)\} \cap (A^{\prime} \cup \bigcup _{w \in A^{\prime}\setminus V(G_i) \,:\, \psi^{\prime}(w) = c}N_G(w))$ . Hence, $D - \textbf{R}_{v, c, i}(A^{\prime}, \psi^{\prime}) \geq D - \textbf{R}_{v, c, i}(A, \psi ) - t$ , and since $I_{A, \psi } \leq 3D = s$ , it follows that $ \textbf{R}_{v, c, i}$ has upward $(s, \delta )$ -certificates, as desired. Therefore by Theorem 3.3 with $t = D^{2/3}$ ,

\begin{equation*} {{\mathbb {P}}\left [\left | \textbf {R}_{v, c, i} - {{\mathbb {E}}\left [ \textbf {R}_{v, c, i}\right ] }\right | \gt D^{2/3}\right ] } \leq 4\exp \left (-\frac {D^{4/3}}{48D}\right ) \leq \exp \left (-D^{1/4}\right ), \end{equation*}

as desired.

Finally, we prove (11). We apply Theorem 3.3 with exceptional outcomes $\boldsymbol{\Omega} ^* = \boldsymbol{\Omega} ^*_{v, i}$ . To that end, we show that $ \textbf{Y}_{v, c, i}$ has upward $(s, \delta )$ -certificates with respect to $ \boldsymbol{\Omega} ^*$ , where $s = 4D$ and $\delta = \log D$ . Let $(A, \psi ) \in \boldsymbol{\Omega} \setminus \boldsymbol{\Omega} ^*$ . For every $u \in \{x \in N_{G_i}(v) \cap A \,:\, c \in L(x)\} \setminus X_{A, \psi }$ , there is at least one neighbour $w \in N_{G}(u)$ of $u$ such that $w \in A$ and $\psi (w) = \psi (u)$ . Choose one such neighbour, and denote it by $w_u$ . Let $I_{A, \psi }$ index the trials determining whether $u \in A$ , whether $w_u \in A$ , and the assignment of $\psi (u)$ and $\psi (w_u)$ for each $u \in \{x \in N_{G_i}(v) \cap A \,:\, c \in L(x)\}$ . Since $(A, \psi )\notin \boldsymbol{\Omega} ^*_v$ , the multi-set $\{w_u \,:\, u \in (N_{G_i}(v) \cap A) \setminus X_{A, \psi },\ c \in L(u)\}$ has maximum multiplicity at most $\log D$ (as otherwise, if $w$ appears more than $\log D$ times in this multi-set, then $ \textbf{F}_{w, \psi (w), i}(A, \psi ) \gt \log D$ ). Hence, if $(A^{\prime}, \psi^{\prime}) \in \boldsymbol{\Omega}$ differs from $(A, \psi )$ in at most $t/ \log D$ of the trials indexed by $I_{A, \psi }$ (and differs arbitrarily for trials not indexed by $I_{A, \psi }$ ), then all but at most $t$ vertices in $\{x \in N_{G_i}(v) \cap A \,:\, c \in L(x)\} \setminus X_{A, \psi }$ are in $\{x \in N_{G_i}(v) \cap A^{\prime} \,:\, c \in L(x)\} \setminus X_{A^{\prime}, \psi^{\prime}}$ . Hence, $ \textbf{Y}_{v, c, i}(A^{\prime}, \psi^{\prime}) \geq \textbf{Y}_{v, c, i}(A, \psi ) - t$ , and since $I_{A, \psi } \leq 4D = s$ , it follows that $ \textbf{Y}_{v, c, i}$ has upward $(s, \delta )$ -certificates, as desired. Therefore by Theorem 3.3 with $t = D^{2/3}$ and Proposition 4.4,

\begin{equation*} {{\mathbb {P}}\left [\left | \textbf {Y}_{v, c, i} - {{\mathbb {E}}\left [ \textbf {Y}_{v, c, i}\right ] }\right | \gt D^{2/3}\right ] } \leq 4\exp \left (-\frac {D^{4/3}}{64D\log ^2 D}\right ) + 44^3C^3D^5\left (\log D\right )^{-\log D}\leq \exp \left (-D^{1/4}\right ), \end{equation*}

as desired.

Finally we can prove Lemma 2.1, using Lemmas 4.3 and 4.5.

Proof of Lemma 2.1 . First, rather than showing (2.1.1), it suffices to show that $|L^{\prime}(v)| \geq \left (1 - p/\Lambda \right )^{DC}\Lambda - \Lambda ^{4/5}$ for every $v \in V(G)\setminus X$ , since we can truncate each list until equality holds.

We consider $(A, \psi )$ chosen randomly as described earlier, and we define the following set of bad events for each vertex $v\in V(G)$ and $c\in L(v)$ and $i \in [m]$ :

\begin{align*} \mathcal A_v &= \left \{ (A, \psi ) \,:\, \boldsymbol{\Lambda}_v(A, \psi ) \lt{{\mathbb{E}}\left [ \boldsymbol{\Lambda}_v\right ] } - \Lambda ^{4/5}\right \},\\ \mathcal A_{v, c, i} &= \left \{ (A, \psi ) \,:\, \textbf{Y}_{v, c, i} \gt{{\mathbb{E}}\left [ \textbf{Y}_{v, c, i}\right ] } + D^{4/5}/2 \right \}, \text{ and}\\ \mathcal A^{\prime}_{v, c, i} &= \left \{(A, \psi ) \,:\, \textbf{R}_{v,c,i}(A, \psi ) \gt{{\mathbb{E}}\left [ \textbf{R}_{v, c, i}\right ] } + D^{4/5}/2\right \}. \end{align*}

Letting $\mathcal A$ be the union of all such bad events, note that each event in $\mathcal A$ is mutually independent of all but at most $(\lceil \Lambda \rceil CD)^4 \leq (11C^2D^2)^4$ other events in $\mathcal A$ . By Lemma 4.5, every event in $\mathcal A$ occurs with probability at most $\exp \left (-D^{1/4}\right )$ . Therefore by the Lovász Local Lemma, there exists $(A, \psi )\notin \mathcal A$ .

Now we show that $X \,:\!=\, X_{A, \psi }$ and $L^{\prime} \,:\!=\, L_{A, \psi }|_{V(G)\setminus X}$ satisfy the lemma with $\phi \,:\!=\, \psi |_X$ . Indeed, $\phi$ is an $L|_X$ -colouring of $G[X]$ and since $X \subseteq A$ , we have $L^{\prime}(v) \subseteq L(v)\setminus \{\phi (u) \,:\, u \in N_G(v) \cap X\}$ , as required. Moreover, for every $v\in V(G)$ , since $(A, \psi ) \notin \mathcal A_v$ , by (7), we have $|L^{\prime}(v)| \geq \left (1 - p/\Lambda \right )^{DC}\Lambda - \Lambda ^{4/5}$ , as desired, and if $v \in V(G_i)$ and $c\in L(v)$ , then since $(A, \psi ) \notin \mathcal A_{v, c, i} \cup \mathcal A^{\prime}_{v, c, i}$ , by (4), (8), and (9), we have (2.1.2), as desired.

5. Correspondence colouring

In this section we introduce correspondence colouring and describe how to generalize Theorem 1.3 to this setting.

For convenience, if $uv\in E(G)$ , $c_1\in L(u)$ , $c_2\in L(v)$ , and $(u, c_1)(v, c_2)\in M(uv)$ , we will just write $c_1c_2\in M(uv)$ . For each $v \in V(G)$ and $c \in L(v)$ , we will let $N_{G, (L, M)}(v, c) \,:\!=\, \{(u, c^{\prime}) \,:\, uv \in E(G), cc^{\prime} \in M(uv)\}$ , and we omit the subscript in $N_{G, (L, M)}(v, c)$ when it is clear from the context. Note that if for each $e=uv\in E(G)$ and $c\in L(u)\cap L(v)$ , we have $cc\in M(uv)$ , then an $(L, M)$ -colouring is an $L$ -colouring. Hence, every graph $G$ satisfies $\chi _\ell (G)\leq \chi _{DP}(G)$ .

For a graph $G$ with correspondence assignment $(L, M)$ , we define the colour degree of each $v\in V(G)$ and $c \in L(v)$ to be $d_{G, (L, M)}(v, c) \,:\!=\, |N_{G, (L, M)}(v, c)|$ and the maximum colour degree to be $\Delta (G, (L, M)) \,:\!=\, \max _{v\in V(G)}\max _{c \in L(v)}d_{G, (L, M)}(v, c)$ . With only minor modifications to the argument used to prove Theorem 1.3, which we describe below, we can strengthen Theorem 1.3 to the setting of correspondence colouring, as follows.

As mentioned, Theorem 5.2 implies that Corollary 1.1 actually holds for the correspondence chromatic number, which in turn implies the main result of [Reference Molloy and Postle19], that linear and uniform hypergraphs of maximum degree at most $D$ have correspondence chromatic index at most $D + o(D)$ .

To prove Theorem 5.2, we use the argument presented in Section 2, but with Lemmas 2.1 and 2.3 replaced with the following lemmas, respectively.

Lemma 5.4, like Lemma 2.3, can be proved with a straightforward application of the Lovász Local Lemma. A stronger result (with ‘ $|L(v)| \geq 8D$ ’ replaced by ‘ $|L(v)| \geq 2D$ ’) also follows easily from a well-known result of Haxell [Reference Haxell11] on independent transversals.

Therefore it remains to describe how the argument presented in Section 4 to prove Lemma 2.1 can be modified to obtain Lemma 5.3. We consider the same probability space $( \boldsymbol{\Omega}, \boldsymbol{\Sigma},{\mathbb{P}})$ of pairs $(A, \psi )$ , but we instead define $L_{A, \psi }(v) \,:\!=\, \{c \in L(v) \,:\, (v, c) \in \left (\{v\}\times L(v)\right ) \setminus \bigcup _{u\in A}N(u, \psi (u))\}$ . The definition of $X_{A, \psi }$ and of $ \boldsymbol{\Lambda}_v(A, \psi )$ remains the same, but we replace the definitions of $ \textbf{D}_{v, c, i}(A, \psi )$ , $ \textbf{Y}_{v, c, i}(A, \psi )$ , and $ \textbf{R}_{v, c, i}(A, \psi )$ with the following:

• • $ \textbf{D}_{v, c, i}(A, \psi ) \,:\!=\, \left |\left \{ (u, c^{\prime}) \in N_{G_i, (L|_{V(G_i)}, M|_{E(G_i)})}(v, c) \,:\, u \notin X_{A, \psi }, c^{\prime} \in L_{A, \psi }(u)\right \}\right |$ ,

• • $ \textbf{Y}_{v, c, i}(A, \psi ) \,:\!=\, \left |\left \{ (u, c^{\prime}) \in N_{G_i, (L|_{V(G_i)}, M|_{E(G_i)})}(v, c) \,:\, u \in A \setminus X_{A, \psi }\right \}\right |$ , and

• • $ \textbf{R}_{v, c, i}(A, \psi ) \,:\!=\, \left |\left \{ (u, c^{\prime}) \in N_{G_i, (L|_{V(G_i)}, M|_{E(G_i)})}(v, c) \,:\, u \notin A\right \} \setminus \bigcup _{u \in A\setminus V(G_i)}N_{G, (L, M)}(u, \psi (u))\right |$ .

The definition of $ \boldsymbol{\Omega} ^*_{v, i}$ remains the same, but we replace the definition of $ \textbf{F}_{v, c, i}(A, \psi )$ with the following:

• • $ \textbf{F}_{v, c, i}(A, \psi ) \,:\!=\, \left |\left \{(u, c^{\prime}) \in N_{G_i, (L_{V(G_i)}, M_{E(G_i)})}(v, c) \,:\, \psi (u) = c^{\prime}\right \}\right |$ .

The remainder of the proof can be obtained via straightforward modifications, so we omit the details. This completes the proof of Lemma 5.3 and in turn implies Theorem 5.2.

6. Proof of Theorems 1.4 and 1.5

In this section, we prove Theorems 1.4 and 1.5. Recall that $f(m,{\mathcal G})$ is the largest possible chromatic number of the union of at most $m$ nearly disjoint graphs in $\mathcal G$ , and ${\mathcal G}^\chi _n$ is the set of graphs of chromatic number at most $n$ . We begin by providing the construction that certifies that $f(3,{\mathcal G}^\chi _n) \geq n + 1$ .

To prove Theorem 1.4, it remains to prove the bound $f(m,{\mathcal G}^\chi _n) \leq m + n - 2$ when $m + n$ is sufficiently large. First we note the following immediate consequence of the main result in [Reference Kang17] (namely that the Erdős–Faber–Lovász conjecture holds for all sufficiently large $n$ ).

Proof of Theorem 1.4(i). Suppose to the contrary, and let $G_1, \dots, G_m$ be nearly disjoint graphs, each of chromatic number at most $n$ , such that $\chi \left(\bigcup _{i=1}^m G_i\right) \gt m + n - 2$ and $|V\left(\bigcup _{i=1}^m G_i\right)|$ is minimum. Note that $n \geq 2$ as otherwise $\chi \left(\bigcup _{i=1}^m G_i\right) = 1$ . By Theorem 6.1 with $n + m - 2$ playing the role of $C$ , there exists $i \in [m]$ such that $|V(G_i)| \gt m + n - 2$ , and we may assume without loss of generality that $i = 1$ . Let $X \,:\!=\, V(G_1) \setminus \bigcup _{i=2}^m V(G_i)$ . Since $G_1, \dots, G_m$ are nearly disjoint,

(13) \begin{equation} |X| \geq |V(G_1)| - (m - 1) \geq n. \end{equation}

Since $\chi (G_1) \leq n$ , there is a partition $I_1, \dots, I_n$ of $V(G_1)$ into independent sets. By (13) and the pigeonhole principle, there exists some $j\in [n]$ such that either $|I_j \cap X| \gt 1$ or both $I_j \cap X$ and $I_j \setminus X$ are nonempty. We may assume without loss of generality that $j = 1$ , so there exist distinct vertices $u, v\in I_1$ such that $u \in X$ . Let $G^{\prime}_1$ be the graph obtained from $G_1$ by identifying $u$ and $v$ into a single new vertex. Crucially, $G^{\prime}_1, G_2, \dots, G_m$ are nearly disjoint, $\chi (G^{\prime}_1) \leq n$ , and $\chi (G^{\prime}_1 \cup \bigcup _{i=2}^m G_i) \geq \chi \left(\bigcup _{i=1}^m G_i\right)$ , contradicting the choice of $G_1, \dots, G_m$ to have $|V\left(\bigcup _{i=1}^m G_i\right)|$ minimum.

Now we prove Theorem 1.5. In this proof, we use a Latin square to construct a graph of large chromatic number. An order- $n$ Latin square is an $n\times n$ array of $n$ symbols where each row and column contains each symbol exactly once. That is, for an order- $n$ Latin square $L$ , with rows, columns, and symbols indexed by $A$ , $B$ , and $C$ , respectively, we have $\{L(a, b) \,:\, b \in{B}\} ={C}$ for every $a \in{A}$ and $\{L(a, b) \,:\, a \in{A}\} ={C}$ for every $b \in{B}$ .

Proof of Theorem 1.5 . First, let $G \cong K_{m}$ , and since $3 \mid m$ , there exist sets ${A},{B},{C} \subseteq V(G)$ of size $m/ 3$ that partition $V(G)$ . Since $n \geq m - 1$ , we can let $G^{\prime}$ be the multigraph obtained from $G$ by adding $n - (m - 1) + m/ 3$ loops to each vertex in ${A} \cup{B}$ and $n - (m - 1)$ loops to each vertex in $C$ , and we let $H$ be the line graph of $G^{\prime}$ . Let $L$ be an order- $(m/ 3)$ Latin square with rows, columns, and symbols indexed by $A$ , $B$ , and $C$ , respectively, and for each $a \in{A}$ and $b \in{B}$ , let $e^{ab}_1$ denote the edge in $H$ with ends $ab$ and $ac$ , and let $e^{ab}_2$ denote the edge in $H$ with ends $ab$ and $bc$ , where $c = L(a, b)$ . Let

\begin{equation*} H^{\prime} \,:\!=\, H - \bigcup _{(a,b) \in {A}\times {B}}\left\{e^{ab}_1, e^{ab}_2\right\}. \end{equation*}

We prove that $H^{\prime}$ has chromatic number at least $n + m/ 6$ and is a nearly disjoint union of $m$ graphs of chromatic number $n$ , as desired.

To that end, for each $v \in V(G)$ , let $G_v\,:\!=\, H^{\prime}[\{e \in E(G^{\prime}) \,:\, e \ni v\}]$ . By construction, the graphs in $\{G_v \,:\, v \in V(G)\}$ are nearly disjoint, and $H^{\prime} = \bigcup _{v \in V(G)} G_v$ . Moreover, $G_v$ is isomorphic to $K_{n + m/3}$ with a matching of size $m/3$ removed for every $v \in{A} \cup{B}$ and is isomorphic to $K_{n}$ for every $v \in{C}$ , so $\chi (G_v) = n$ for every $v \in V(G)$ , as required.

To prove that $\chi (H^{\prime}) \geq n + m/ 6$ , we let $\phi$ be a proper colouring of $H^{\prime}$ using at most $k$ colours, and we show that $k \geq n + m/6$ , as follows. First, let

\begin{align*}{B}_a &\,:\!=\, \{b \in{B} \,:\, \phi (ab) = \phi (ac),\ \text{where } c = L(a, b)\} \text{ for every $a \in{A}$} && \mathrm{and}\\{A}_b &\,:\!=\, \{a \in{A} \,:\, \phi (ab) = \phi (bc),\ \text{where } c = L(a, b)\} \text{ for every $b \in{B}$}. \end{align*}

We claim that the following holds:

1. (a) $|{B}_a| \geq n + m/ 3 - k$ for every $a \in{A}$ ,

2. (b) $|{A}_b| \geq n + m/ 3 - k$ for every $b \in{B}$ , and

3. (c) $\sum _{a\in{A}}|{B}_a| + \sum _{b \in{B}}|{A}_b| \leq \frac{m^2}{9}$ .

Altogether, a)–(c) imply that $(2m/ 3)(n + m/ 3 - k) \leq m^2/ 9$ . Rearranging terms in this inequality, we have $k \geq n + m/ 6$ , as desired. Thus, it remains to prove (c)–(c).

To prove (a), fix some $a \in{A}$ . Since $\phi$ is a proper colouring, the colours assigned to $\{ab \,:\, b \in{B}_a\}$ and the colours assigned to vertices of $G_a$ not incident to an edge of $\left\{e^{ab}_1 \,:\, b \in{B}_a\right\}$ are distinct. Thus, since at most $k$ colours are used in total, $|{B}_a| + n + m/ 3 - 2|{B}_a| \leq k$ . Rearranging terms in this inequality, we have $|{B}_a| \geq n + m/ 3 - k$ , as desired. The proof of (b) is essentially the same, so we omit the details.

To prove (c), suppose to the contrary that $\sum _{a\in A}|{B}_a| + \sum _{b\in B}|{A}_b| \gt m^2/ 9$ . In this case, there is a pair $(a, b) \in{A} \times{B}$ such that $a \in{A}_b$ and $b \in{B}_a$ . Letting $c = L(a, b)$ , since $a \in{A}_b$ , we have $\phi (ab) = \phi (bc)$ , and since $b \in{A}_a$ , we have $\phi (ab) = \phi (ac)$ . However, since $\phi$ is a proper colouring, $\phi (bc) \neq \phi (ac)$ , a contradiction.