Asymmetric list sizes in bipartite graphs

Given a bipartite graph with parts $A$ and $B$ having maximum degrees at most $\Delta_A$ and $\Delta_B$, respectively, consider a list assignment such that every vertex in $A$ or $B$ is given a list of colours of size $k_A$ or $k_B$, respectively. We prove some general sufficient conditions in terms of $\Delta_A$, $\Delta_B$, $k_A$, $k_B$ to be guaranteed a proper colouring such that each vertex is coloured using only a colour from its list. These are asymptotically nearly sharp in the very asymmetric cases. We establish one sufficient condition in particular, where $\Delta_A=\Delta_B=\Delta$, $k_A=\log \Delta$ and $k_B=(1+o(1))\Delta/\log\Delta$ as $\Delta\to\infty$. This amounts to partial progress towards a conjecture from 1998 of Krivelevich and the first author. We also derive some necessary conditions through an intriguing connection between the complete case and the extremal size of approximate Steiner systems. We show that for complete bipartite graphs these conditions are asymptotically nearly sharp in a large part of the parameter space. This has provoked the following. In the setup above, we conjecture that a proper list colouring is always guaranteed * if $k_A \ge \Delta_A^\varepsilon$ and $k_B \ge \Delta_B^\varepsilon$ for any $\varepsilon>0$ provided $\Delta_A$ and $\Delta_B$ are large enough; * if $k_A \ge C \log\Delta_B$ and $k_B \ge C \log\Delta_A$ for some absolute constant $C>1$; or * if $\Delta_A=\Delta_B = \Delta$ and $ k_B \ge C (\Delta/\log\Delta)^{1/k_A}\log \Delta$ for some absolute constant $C>0$. These are asymmetric generalisations of the above-mentioned conjecture of Krivelevich and the first author, and if true are close to best possible. Our general sufficient conditions provide partial progress towards these conjectures.


Introduction
List colouring of graphs, whereby arbitrary restrictions on the possible colours used per vertex are imposed, was introduced independently by Erdős, Rubin and Taylor [10] and by Vizing [16]. Let G = (V, E) be a simple, undirected graph. For a positive integer k, a mapping L : V → Z + k is called a k-list-assignment of G; a colouring c : V → Z + is called an L-colouring if c(v) ∈ L(v) for any v ∈ V . We say G is k-choosable if for any k-list-assignment L of G there is a proper L-colouring of G. The choosability ch(G) (or choice number or list chromatic number) of G is the least k such that G is k-choosable.
Note that if G is k-choosable then it is properly k-colourable. On the other hand, Erdős, Rubin and Taylor [10] exhibited bipartite graphs with arbitrarily large choosability. Let K n 1 ,n 2 denote a complete bipartite graph with part sizes n 1 and n 2 . More succinctly Theorem 1 says that ch(K n,n ) ∼ log 2 n as n → ∞.
There are two contrasting ways to try to strengthen this last statement. First does the lower bound hold more generally; that is, does a bipartite graph with minimum degree δ have choosability Ω(log δ) as δ → ∞? Indeed this was shown by the first author [1] using a probabilistic argument; later Saxton and Thomason [15] proved the asymptotically optimal lower bound of (1 + o(1)) log 2 δ. Second does the upper bound hold more generally; that is, does a bipartite graph with maximum degree ∆ always have choosability O(log ∆) as ∆ → ∞? Krivelevich and the first author conjectured this in 1998.

Conjecture 2 ([2]
). There is some absolute constant C > 0 such that any bipartite graph of maximum degree at most ∆ is k-choosable if k ≥ C log ∆.
Johansson's result for triangle-free graphs [12] gives the conclusion with k ≥ C∆/ log ∆, which is far from the conjectured bound. Conjecture 2 is an elegant problem in a wellstudied area, but apart from constant factors 1 there has been no progress until now.
Despite the apparent difficulty of Conjecture 2, we propose an asymmetric refinement. Given a bipartite graph G = (V =A ∪ B, E) with parts A, B and positive integers k A , k B , a mapping L : A → Z + k A , B → Z + k B is called a (k A , k B )-list-assignment of G. We say G is (k A , k B )-choosable if there is guaranteed a proper L-colouring of G for any such L. Problem 3. Given ∆ A and ∆ B , what are optimal choices of k A ≤ ∆ A and k B ≤ ∆ B for which any bipartite graph G = (V =A ∪ B, E) with parts A and B having maximum degrees at most ∆ A and ∆ B , respectively, is (k A , k B )-choosable?
We have the upper bounds on k A , k B , since the problem is trivial if k A > ∆ A or k B > ∆ B .
Note that Problem 3, since it has a higher-dimensional parameter space, has wider scope than Conjecture 2 and is necessarily more difficult. However, the extra generality in Problem 3 has permitted a glimpse at an unexpected and basic connection between list colouring and combinatorial design theory (Theorem 6). This has prompted us to explore specific areas of the parameter space in Problem 3, motivating concrete versions of Problem 3 in the spirit of Conjecture 2 (Conjecture 7). One hope of ours is that further study of these problems may yield insights into Conjecture 2. But in fact, already in the present work, we have obtained asymmetric progress towards Conjecture 2 (Corollary 10).
Our first main result provides general progress towards Problem 3.
satisfy one of the following conditions.
Then any bipartite graph G = (V =A ∪ B, E) with parts A and B having maximum degrees at most ∆ A and ∆ B , respectively, is (k A , k B )-choosable.
Theorem 4 under condition (i) follows from a simple application of the Lovász Local Lemma, as we show in Section 2. This sufficient condition for Problem 3 is related to independent transversals in hypergraphs, cf. [8,11], and to single-conflict chromatic number [7]. In the most asymmetric settings (when k A and ∆ A are fixed constants), condition (i) is sharp up to a constant factor. We prove Theorem 4 under condition (ii) in Section 3 with the Lovász Local Lemma and a link to the coupon collector problem. In an attempt to clear up the 'parameter soup' arising from Problem 3 and the above sufficient conditions, we will discuss specific natural context for these after presenting some further results. We complement our sufficient conditions for (k A , k B )-choosability with necessary ones, given mainly by the complete bipartite graphs. An easy boundary case is provided as follows, for warmup. This generalises a classic non-k-choosable construction.
Our second main result is a related but much broader necessary condition for Problem 3, also via the complete case. Let us write M (k 1 , k 2 , ℓ) for the hypergraph Turán number defined as the minimum number of edges in a k 2 -uniform hypergraph on ℓ vertices with no independent set of size ℓ − k 1 . This parameter is equivalent to the extremal size of approximate Steiner systems; in particular, M(k 1 , k 2 , ℓ) is equal to the cardinality of a smallest k 1 -(ℓ, ℓ − k 2 , Z + ) design, cf. [9,Ch. 13] and [13]. (Recall that a t-(v, k, Z + ) design is a collection of k-element subsets, called blocks, of some v-element set X such that each t-element subset of X is contained in at least one block.) We can draw a link between this classical extremal parameter and Problem 3. We show the following result in Section 4.
This link allows us, using known results for M, to read off decent necessary conditions for specific parameterisations of Problem 3. As one example, a lower bound on ch(K n,n ) of the form ch(K n,n ) 1 2 log 2 n follows from Theorem 6 with the choice k 1 = k 2 = k − 1, k A = k B = k, and ℓ = 2k − 1 for some k. As another example, a slightly weaker form of the necessary condition of Proposition 5 follows with the choice We detail both of these easy examples in Section 4.
Note though that these last two examples show that Theorem 6 provides suboptimal necessary conditions for (k A , k B )-choosability even in the complete case. Furthermore, in Section 8 we give a construction to show that the complete case cannot, in general, be precisely extremal for Problem 3. Nevertheless, we surmise that Theorem 6 provides some good rough borders for Problem 3. More specifically, in Section 5 we give some basic sufficient conditions for (k A , k B )-choosability specific to the complete case and show in Section 6, through some routine asymptotic calculus, how these roughly match with Theorem 6 over a broad family of parameterisations for Problem 3. Then, just as Conjecture 2 was informed by Theorem 1, the asymptotic behaviour of (k A , k B )-choosability in the complete case leads us to conjecture the following concrete versions of Problem 3.

Conjecture 7.
Let the positive integers ∆ A , ∆ B , k A , k B satisfy one of the following.
We note that in Section 5 we also show how this conjecture holds for the complete bipartite graphs (Theorem 15). Conjecture 7 constitutes three natural asymmetric analogues of Conjecture 2. The first is weaker than Conjecture 2; the latter two are stronger. We discuss these separately in turn. Behind condition (i), Conjecture 7 concerns the question, for what positive functions f does k A ≥ f (∆ A ) and k B ≥ f (∆ B ) suffice for (k A , k B )-choosability, no matter how far apart ∆ A and ∆ B are? Essentially we posit that some f with f (x) = x o(1) as x → ∞ will work, and this would be best possible due to the complete case (Theorem 16(i)). In particular, f (x) = O(log x) is impossible, in contrast to Conjecture 2. But in fact, Theorem 4 implies that Conjecture 7 under condition (i) reduces to its most symmetric form.

Proof. It suffices to check condition (i) of Theorem 4 with k
Assume condition (i) of Conjecture 7, and moreover assume the truth of the conjecture only for its most symmetric form ∆ A = ∆ B = ∆. We may assume that ∆ 0 > (4/ε) 1/ε , and so, without loss of generality, we may also assume by the first part that ∆ A ≤ ∆ B ≤ ∆ 2/ε A . For any bipartite graph G = (V =A ∪ B, E) with parts A and B having maximum degrees at most ∆ A and ∆ B , we have from the assumption that G is k-choosable for, say, k = ∆ ε 3 B ≤ ∆ 2ε 2 A , provided ∆ B , and thus ∆ A , is large enough as a function of ε. This is smaller than ∆ ε A for ∆ A sufficiently large as a function of ε, as required. Thus Conjecture 7 under condition (i) would follow from a weaker form of Conjecture 2. By an earlier work due to Davies, de Joannis de Verclos, Pirot and the third author [5], we so far know that f (x) = (1 + o(1))x/ log x works.
Under condition (ii), Conjecture 7 concerns a 'crossed' version of the previous question, so for what positive functions g does k A ≥ g(∆ B ) and k B ≥ g(∆ A ) suffice for (k A , k B )choosability? In this case, we conjecture that g(x) = O(log x) will work, which coincides with Conjecture 2 in the symmetric case ∆ A = ∆ B = ∆. The complete bipartite graphs demonstrate the hypothetical sharpness of this assertion up to a constant factor for nearly the entire range of possibilities for ∆ A and ∆ B (Theorem 16(ii)). Some modest partial progress towards Conjecture 7 under condition (ii) follows from Theorem 4. Corollary 9. Given ε > 0, there exists δ such that for ∆ B large enough, any bipartite graph G = (V =A ∪ B, E) with parts A and B having maximum degrees at most ∆ A and Proof. This follows for ∆ B large enough from condition (i) of Theorem 4 with either Under condition (iii), Conjecture 7 concerns the setting most closely related to Conjecture 2. It suggests how Problem 3 might behave for the symmetric case ∆ A = ∆ B = ∆. The complete bipartite graphs demonstrate its hypothetical sharpness up to a constant factor for the entire range of possibilities for k A , and thus symmetrically k B (Theorem 16(iii)). Theorem 4 provides the following partial progress towards Conjecture 7 under condition (iii). In fact, this constitutes significant (asymmetric) progress towards Conjecture 2, and is a first concrete step in this longstanding problem.
Proof. This follows for ∆ large enough from condition (ii) in Theorem 4 with ∆ A = ∆ B = ∆ and either k A = (1+ε)∆/ log 4 ∆ and k B = 2 or k A = (1+ε)∆/ log ∆ and k B = log ∆. In Section 7 we are able to nearly completely characterise (k A , k B )-choosability for complete bipartite graphs if k A ≥ ∆ A − 1. In Section 9, we give a connection between minimum degree and (k A , k B )-choosability along the lines of [1].

Probabilistic preliminaries
We will use the following standard probabilistic tools, cf. [3].
A Chernoff Bound. For X ∼ Bin(n, p) and ε ∈ [0, 1], P(X < (1 − ε)np) ≤ exp(− ε 2 2 np). The Lovász Local Lemma. Consider a set E of (bad) events such that for each A ∈ E (i) P(A) ≤ p < 1, and (ii) A is mutually independent of a set of all but at most d of the other events.
If ep(d + 1) ≤ 1, then with positive probability none of the events in E occur.

A sufficient condition via transversals
In this section, we prove Theorem 4 under condition (i). We prefer to state and prove a slightly stronger form. To do so, we need some notational setup. Let H = (V, E) be a hypergraph. The degree of a vertex in H is the number of edges containing it. Given some partition of V , a transversal of H is a subset of V that intersects each part in exactly one vertex. A transversal of H is called independent if it contains no edge, cf. [8].
Lemma 11. Fix k ≥ 2. Let H be a k-uniform vertex-partitioned hypergraph, each part being of size ℓ, such that every part has degree sum at most ∆. If ℓ k ≥ e(k(∆ − 1) + 1), then H has an independent transversal.

Proof of Theorem 4 under condition
We would like to show that there is a proper L-colouring of G. We do so by defining a suitable hypergraph Note that H is a k A -uniform vertex-partitioned hypergraph, where the parts are naturally induced by each list in B and so are each of size k B . We have defined H and its partition so that any independent transversal corresponds to a special partial L-colouring of G. In particular, it is an L-colouring of the vertices in B for which there is guaranteed to be a colour in L(v) available for every v ∈ A, and so it can be extended to a proper L-colouring of all G.
Every part in H has degree sum at most ∆ B Lemma 11. Write H = (V, E) and suppose ℓ k ≥ e(k(∆ − 1) + 1). Consider the random transversal T formed by choosing one vertex from each part independently and uniformly. For Since by assumption e(1/ℓ k )(k(∆ − 1) + 1) ≤ 1, there is a positive probability that T is independent by the Lovász Local Lemma.

A sufficient condition via coupon collection
Proof of Theorem 4 under condition (ii). Let k A , k B , ∆ A , ∆ B satisfy condition (ii). Let L be a (k A , k B )-list-assignment of G. We would like to show that there is a proper L-colouring of G. To this end, colour each vertex w ∈ B, randomly and independently, by a colour chosen uniformly from its list L(w). Let T v,c be the event that v ∈ A has a neighbour coloured with colour c. Let T v be the event that T v,c happens for all c ∈ L(v). The proof hinges on the following claim, which is related to the coupon collector problem, cf. e.g. [6].
Claim. The events T v,c , for fixed v as c ranges over all colours in L(v), are negatively correlated. In particular, Proof. We have to prove, for every I ⊂ L(v), that P(∀c ∈ I : T v,c ) ≤ c∈I P(T v,c ). If some colour c ∈ L(v) is not in the list of any neighbour of v, both sides equal zero and so the inequality holds. So assume this is not the case. We prove the statement by induction on |I|. When |I| ≤ 1 the statement is trivially true. Let I ⊂ L(v) be a subset for which the statement is true and let c ′ ∈ L(v) \ I. We now prove the statement for I ′ = I ∪ {c ′ }. We have P(∀c ∈ I : T v,c ) ≤ P(∀c ∈ I : T v,c | ¬T v,c ′ ) as the probability to use a colour in I is larger if in all neighbouring lists the colour c ′ is removed. This is equivalent to This last expression is at most c∈I ′ P(T v,c ) by the induction hypothesis, as desired. ♦ is concave and increasing, Jensen's Inequality applied with the claim implies that Each event T v is mutually independent of all other events T u apart from those corresponding to vertices u ∈ A that have a common neighbour with v in G. As there are at most ∆ A (∆ B − 1) such vertices besides v, the Lovász Local Lemma guarantees with positive probability that none of the events T v occur, i.e. there is a proper L-colouring, as desired.

The complete case and Steiner systems
In this section, we investigate general necessary conditions for (k A , k B )-choosability via the complete bipartite graphs. Inspired in part by Theorem 1 and related work of Bonamy and the third author [4], this leads naturally to the study of an extremal set theoretic parameter. For positive integers k 1 , k 2 , ℓ, we say that a family F of k 2 -element subsets of [ℓ] has Property A(k 1 , k 2 , ℓ) (A is for asymmetric) if there is a k 1 -element subset of [ℓ] that intersects every set in F . We then define M (k 1 , k 2 , ℓ) to be the cardinality of a smallest family of k 2 -element subsets of [ℓ] that does not have Property A(k 1 , k 2 , ℓ). Note that this definition of M coincides with the definition given before the statement of Theorem 6.
Proof of Theorem 6. We define a (k A , k B )-list-assignment L as follows. Let F 1 be a family of M (k 1 , k A , ℓ) k A -element subsets of [ℓ] without Property A(k 1 , k A , ℓ). Let F 2 be a family of M (k 2 , k B , ℓ) k B -element subsets of [ℓ] without Property A(k 2 , k B , ℓ). Assign the sets of F 1 as lists to the vertices in A and the sets of F 2 as lists to the vertices in B. Suppose that c is an L-colouring. Then C 1 = {c(a) | a ∈ A} intersects every set in F 1 and so |C 1 | ≥ k 1 + 1 by assumption and similarly C 2 = {c(b) | b ∈ B} has cardinality |C 2 | ≥ k 2 + 1. So |C 1 | + |C 2 | ≥ k 1 + k 2 + 2 > ℓ, implying that c cannot be a proper colouring.
Since M (ℓ − k A , k A , ℓ) = ℓ k A , we have the following corollary of Theorem 6.
Although they are trivial, the following observations already show that Theorem 6 and Corollary 12 cannot be improved much in general.
This implies ch(K n,n ) 1 2 log 2 n, which one can compare to the bounds of Theorem 1.
This produces a necessary condition on k B for (k A , k B )choosability only slightly weaker than Proposition 5.
We present several more instances where Theorem 6 and Corollary 12 are nearly sharp in Section 6. For this, we will have use for the following estimates for the parameter M . A version of this result can be found in [9,Ch. 13], but for completeness we present a standard derivation in the appendix.

Sufficient conditions in the complete case
In this section, we give general sufficient conditions for (k A , k B )-choosability of complete bipartite graphs K a,b . Our strategy for establishing (k A , k B )-choosability in this setting is to take a random bipartition of the set of all colours and try to use one part for colouring A and the other part for colouring B. This yields the following lemma.
Lemma 14. Let the reals 0 < ε, p < 1 and positive integers a, b, k A , k B satisfy either Then the complete bipartite graph G = (V =A ∪ B, E) with |A| = a and |B| = b is (k A , k B )choosable.
Proof. Let L be (k A , k B )-list-assignment of G. Let U = ∪ v∈V L(v), i.e. U is the union of all colour lists. Define a random partition of U into parts L A and L B as follows. For each colour in U, randomly and independently assign it to L B with probability p and otherwise assign it to L A . First assume (1). For a given vertex v ∈ A, the probability that L(v) ∩ L A = ∅ is p k A . For a given vertex v ∈ B, the probability that L(v) ∩ L B = ∅ is (1 − p) k B . So the expected number of vertices v which cannot be coloured with the corresponding list L A or L B is equal to ap k A + b(1 − p) k B . So the probabilistic method then guarantees some choice of the parts L A and L B such that every vertex in A can be coloured with a colour from L A and every vertex in B with a colour from L B .
Otherwise assume (2). For a given vertex w ∈ B, the random variable |L(w) ∩ L B | has a binomial distribution of parameters k B and p. So by a Chernoff bound the probability that |L(w) ∩ L B | is smaller than (1 − ε)k B p is smaller than exp (−ε 2 k B p/2). Thus, with probability greater than 1 For a given vertex v ∈ A, the probability that L(v) ∩ L A = ∅ is p k A . So the expected number of vertices v for which this holds is p k A |A|. Thus by Markov's inequality, the probability that there are at least (1 − ε)k B p such vertices v is smaller than p k A −1 a (1−ε)k B . Due to (2), the probabilistic method guarantees a fixed choice of partition of U into parts L A and L B such that |L(w) ∩ L B | ≥ (1 − ε)k B p for every w ∈ B and the number of vertices v ∈ A such that L(v) ∩ L A = ∅ is smaller than (1 − ε)k B p. Colour any vertex v ∈ A having L(v) ∩ L A = ∅ with an arbitrary colour from L(v). Colour any other vertex v ′ ∈ A with a colour from L(v ′ ) ∩ L A . Finally colour any vertex w ∈ B with some colour in L(w) ∩ L B unused by any vertex of A -this is possible as there were fewer than (1 − ε)k B p colours of L B used to colour vertices in A and the lists in B all have at least (1 − ε)k B p colours from L B .
In either case, we are guaranteed a proper L-colouring of G, as promised.

Asymptotic sharpness in the complete case
We next use the results of Sections 4 and 5 to roughly settle the behaviour of complete bipartite graphs with respect to Problem 3 in several regimes. The conditions in Theorems 15 and 16 naturally correspond to the conditions in Conjecture 7.
Theorem 15. Let the positive integers a, b, k A , k B satisfy one of the following.
The conditions are also sharp in some sense. (i) Given a monotone real function g satisfying g(1) ≥ 1 and g(x) = ω(1) as x → ∞, (ii) For any integer t ≥ 4, either (iii) For any fixed integer k > 1 and ∆ sufficiently large, ∆ A = ∆ B = ∆, k A = k and k B = c(∆/ log ∆) 1/k log ∆ for some constant c = c(k).
Proof. Consider condition (i). Take δ large enough so that g(δ a ) > a. Let k A = δ and b = δ a and note that b The result then follows from Proposition 5. − log log a).

Consider condition (ii). For the former case, let
Here we used the estimation For the latter case, choose ℓ = 2 t + 1 t + log 2 t + 3 log 2 a, k A = 2 t log 2 a and k B = log 2 b − log 2 log a t + log 2 t + 3 .
In either case, the result follows from Corollary 12 and Theorem 13. Note that as ∆ → ∞ and thus ∆ A , ∆ B ≤ ∆ for all ∆ large enough.
We define a (k A , k B )-list-assignment L as follows. Let F be a family of |A| (m/2)element subsets of [m] without Property A((log m)/2, m/2, m) and assign the sets of F as lists to the vertices in A. So there is no ((log m)/2)-element subset of [m] that intersects every list in A. For B, arbitrarily partition [m] into b log ∆ segments of nearly equal size, and assign as lists to the vertices in B all possible k-element subsets chosen from within a single segment.
Note that for any L-colouring, ∪ w∈B L(w) intersects at least one colour from each kelement subset of a segment, and so ∪ w∈B L(w) avoids at most (k − 1)b log ∆ = (log m)/2 colours of [m]. However, as noted above, ∪ v∈A L(v) must have more than (log m)/2 colours of [m], and this precludes a proper L-colouring.

Sharpness in a boundary complete case
In this section, we precisely solve Problem 3 for complete bipartite graphs when k A ≥ ∆ A − 1. When k A ≥ ∆ A + 1, we know that any bipartite G = (A ∪ B, E) must be (k A , k B )choosable. The case k A = ∆ A is handled by Proposition 5 and the fact that its conclusion fails if |A| < δ k or |B| < k. The remainder of this section is devoted to the case k A = ∆ A −1.
For this it will be useful to have the following simple lemma at our disposal. Given a family F of disjoint subsets of X, a transversal of F is a subset of X that intersects every set in F exactly once. An almost-transversal of F is a subset of X that intersects all but one set in F exactly once.
Lemma 17. Suppose F consists of b disjoint sets F 1 , F 2 , . . . , F b such that |F 1 | ≥ |F 2 | ≥ . . . ≥ |F b |. Then every almost-transversal of F is subset of at most |F 1 | transversals of F . As a corollary, if F ⋆ is a family of almost-transversals of F such that every transversal of F contains some element of F ⋆ , then F ⋆ must contain at least |F 2 | · · · |F b | elements.
Proof. Any almost-transversal of F can be extended to a transversal of F by adding exactly one element from F j for some j ∈ [b], and so there are |F j | ≤ |F 1 | choices for this. The corollary follows directly by a union bound argument.
Proof. We write B = {v 1 , v 2 , . . . , v b }. Let L be a (b − 1, δ)-list-assignment of G. If some colour is in the list of at least three different vertices of B, or two colours are both in the lists of two disjoint pairs of vertices of B, then we can certainly L-colour G. (The L-colouring uses the common colour(s) for the respective vertices and an arbitrary L-colouring for the remaining vertices of B, followed by a greedy L-colouring of the vertices in A.) Let L 12 = L(v 1 ) ∩ L(v 2 ) and analogously define L ij for each 1 ≤ i < j ≤ b and let ℓ ij = |L ij |. Consider the family of sets induced by the index pairs of the non-empty L ij . This must form an intersecting family, for otherwise we have an L-colouring of G similarly as above. We consider two cases depending on this family being trivial or not.
(i) (non-trivial family) Without loss of generality, we may assume that only L 12 , L 13 and L 23 are non-empty among all L ij . If there are b − 1 colours which do not occur as the list of a vertex of A, such that the vertices of B can coloured with these b − 1 colours, then G is L-colourable. So if G is not L-colourable, every such collection of b − 1 colours has to occur as the list of a vertex in A. There are This implies that G is L-colourable if By the AM-GM Inequality, the minimum of the last expression subject to a fixed sum ℓ 12 + ℓ 13 + ℓ 23 occurs when ℓ 12 = ℓ 13 = ℓ 23 , and so the minimum is attained when ℓ 12 = ℓ 23 = ℓ 13 = δ/4, leading to a bound of 11 16 δ b−1 . (ii) (trivial family) Without loss of generality, we assume only some of the L 1j are nonempty and ℓ 1b = min{ℓ 1j | 2 ≤ j ≤ b}. There are possible ways to choose one colour from every L(v i ), 2 ≤ i ≤ b, in such a way that L(v 1 ) does contain at least one of these chosen colours. So if A would not contain these combinations among the lists L(a), a ∈ A, one would be able to colour G with these b − 1 colours.
By Lemma 17, one needs also lists L(a), a ∈ A, to make it impossible to colour G by first colouring the vertices B solely with colours not appearing among the L 1j . Hence the number of disjoint lists L(a) to make a list-colouring of G impossible needs to be at least This expression is minimised when δ − b−1 j=2 ℓ 1j and every ℓ 1j for 2 ≤ j ≤ b − 1 are equal to a number of the form ⌊ δ b−1 ⌉, as by an integral version of the AM-GM Inequality the subtracted product attains its maximum when all factors (which sum to a fixed amount) differ by at most 1. When b ≪ δ and b ≥ 5, we have that this is approximately (1 − (1 − 1/(b − 1)) b−1 )δ b−1 , which is smaller than the value 11 16 δ b−1 obtained in the previous case. Equality can be attained, i.e. G is not

Sharper than complete bipartite
In this section, we prove that complete bipartite graphs are not exactly extremal for Problem 3. The complete bipartite graph K δ k −1,k is (k, δ)-choosable, but there are bipartite graphs with ∆ A = k and ∆ B smaller than δ k − 1 which are not (k, δ)-choosable.
Proposition 20. For any δ, k ≥ 2, there is a bipartite graph G = (V = A ∪ B, E) with parts A and B having maximum degrees k and f (δ, k) < δ k , respectively, that is not (k, δ)- Proof. We recursively construct bipartite graphs G i = (A i ∪ B i , E i ) with parts A i and B i having maximum degree k and i j=1 (δ − j + 1) k−1 . We simultaneously define a (k, δ)-listassignment L i of G i such that there is some vertex b i ∈ B i which can only be given one of δ − i colours out of its list in any proper L i -colouring.
Let G 1 be the complete bipartite graph K δ k−1 ,k , and write B 1 = {v 1 , . . . , v k }. For the vertices of B 1 , we assign k disjoint lists of length δ, specifically, L(v j ) = {(j−1)δ+1, . . . , jδ} for j ∈ [k]. For the vertices of A 1 , we assign as lists all possible k-tuples drawn from {1} × k j=2 L(v j ). Since b 1 := v 1 cannot be given the colour 1 in any proper L 1 -colouring, the conditions are satisfied for i = 1.
For the recursion, assume i ≥ 1 and take the disjoint union of k copies of G i , relabelling their (k, δ)-list-assignments so that their colour palettes are mutually disjoint. So the parts A i+1 , B i+1 of the bipartition so far include the disjoint unions of the k respective parts. Let v 1 , . . . , v k be the k copies of b i , and for each j ∈ [k] write L ′ (v j ) for the set of δ − i colours to which the colour of v j is restricted by assumption. By relabelling, we may assume 1 ∈ L ′ (v 1 ). We now add (δ − i) k−1 new vertices to A i+1 that are adjacent to every v j . For these new vertices, we assign as lists all possible k-tuples drawn from {1} and L ′ (v j ) for 2 ≤ j ≤ k. This completes the definition of G i+1 and L i+1 . By induction, b i+1 := v 1 may only be given a colour from L ′ (v 1 ) \ {1} in any proper L i+1 -colouring, and moreover b i+1 is of maximum degree in B i+1 . So (G i+1 , L i+1 , b i+1 ) satisfies the required conditions. This completes the recursive step.
The graph G := G δ with parts A = A δ and B = B δ is not (k, δ)-choosable, since by construction we may not give any colour to b δ in any proper L δ -colouring. Furthermore, the maximum degrees in A and B are respectively k and δ i=1 (δ − i + 1) k−1 = δ i=1 i k−1 , as required.

Degrees and (k A , k B )-choosability
In this section, we give a condition on the minimum degree for concluding that a bipartite graph is not (k A , k B )-choosable. This is a reduction to the behaviour for complete bipartite graphs.
Theorem 21. Suppose the complete bipartite graph G 0 = (V =A 0 ∪ B 0 , E) with |A 0 | = a and |B 0 | = b is not (k A , k B )-choosable. Then any bipartite graph G = (V =A ∪ B, E) with parts A and B such that |A| ≤ |B| and B has minimum degree δ B > 4ab log 4a log k A is not (k A , k B )-choosable.
Proof. Let F a and F b be the collections of lists of sizes a and b, respectively, that can be assigned to A 0 and B 0 , respectively, to certify non-(k A , k B )-choosability of G 0 . Let p = 1/(4b log k A ). Randomly choose X ⊂ A, each vertex included independently with probability p. Then E(|X|) = p|A| and so by Markov's inequality, Define a list-assignment L X of X, by assigning to every vertex of X uniformly and independently a list of F a . Call a vertex v in B good if every member of F a occurs as a list on a neighbour (in X) of v. For any F ∈ F a , let us say that v is not good due to F if F does not as occur as a list on a neighbour of v. Note that So by Markov's inequality, By the probabilistic method, there is some X ⊂ A and a list-assignment L X of X such that |X| ≤ 2p|A| and there are at least |B|/2 good vertices. Fix this choice and let B * be the set of good vertices.
Fix an arbitrary L X -colouring c X of X. There are at most k |X| A possibilities for the colouring c X . Define a list-assignment L B * of B * , by assigning to every vertex of B * uniformly and independently a list of F b . Since every v ∈ B * is good, all lists of F a occur in the neighbourhood of v and at least one choice of a list in F b would imply that v cannot be properly coloured with a colour of that list. Hence the probability that every v ∈ B * can be properly L B * -coloured in agreement with c X is at most The probability that some proper colouring of G can be completed given any L X -colouring c X is smaller than Thus by the probabilistic method there exists a list-assignment L B * of B * such that no proper L B * -colouring can be found in agreement with any L X -colouring.

Conclusion
We have begun the investigation of an asymmetric form of list colouring for bipartite graphs. In one direction, we have found good general sufficient conditions through connections to independent transversals and to the coupon collector problem. This has incidentally yielded a non-trivial advance towards a difficult conjecture of Krivelevich and the first author. In another direction, we have established broad necessary conditions through an unexpected link between the bipartite choosability of complete bipartite graphs and a classic extremal set theoretic or design theoretic parameter. This link has fed naturally into the formulation of three attractive conjectures along these lines. Because of the rich connections this problem has to other important areas of combinatorial mathematics, we are hopeful that further study will lead to novel insights. We remark that Conjecture 7 comprises three asymptotic parameterisations of Problem 3 that we found most natural and interesting, all derived essentially from Theorem 6. There could be several other nice choices. Because the terrain is new, there are many interesting angles we have not yet had the opportunity to fully explore. One possibility, based on the connection to combinatorial design theory, comes to mind. We have that M (2, q 2 , q 2 + q + 1) = q 2 + q + 1 for every prime power q due to the finite projective planes. With a small modification of the substitution of this fact into Corollary 12, we obtain that the complete bipartite graph K 1 6 q 3 (q+1)(q 2 +q+1),q 2 +q+1 is not (3, q 2 )-choosable. On the other hand, Lemma 14 shows this is not that far from optimal, and in particular (2) shows that K q 6 /(80 log q) 2 ,q 2 +q+1 is (3, q 2 )-choosable. It would be interesting to narrow the gap. For the specific case q = 2, a quick computer search checks that K 20,7 is not (3, 4)-choosable, but finding the largest r such that K r,7 is (3, 4)-choosable seems difficult.
Next we prove the upper bound. Fix C a k 1 -element subset of [ℓ]. Let F be a k 2 -element subset of [ℓ] chosen uniformly at random. Then Let F = {F 1 , . . . , F m } be a family of m k 2 -element subsets of [ℓ] chosen uniformly at random. Based on the above calculation, There are ℓ k 1 choices for C, so we have P(F contains a C certifying Property A(k 1 , k 2 , ℓ)) This last expression is less than 1 if which establishes the upper bound.