An Easy Subexponential Bound for Online Chain Partitioning

Bosek and Krawczyk exhibited an online algorithm for partitioning an online poset of width w into w14 lgw chains. We improve this to w6.5 lgw+7 with a simpler and shorter proof by combining the work of Bosek & Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of ladder-free posets. We also provide examples illustrating the limits of our approach. Mathematics Subject Classifications: 05C15, 06A07, 68W27, 68R05 ∗Research of these authors is supported by Polish National Science Center (NCN) grant number 2011/03/B/ST6/01367. †Research of this author was supported in part by NSF grant DMS-0901520 the electronic journal of combinatorics 25(2) (2018), #P2.28 1


Introduction
An online poset P ≺ is a triple (V, P , ≺), where P = (V, P ) is a poset and ≺ is a total order on V , called the presentation order of P .Let P v i be induced by the first i vertices v 1 ≺ • • • ≺ v i .An online chain partitioning algorithm is a deterministic algorithm A that assigns the vertices v 1 ≺ • • • ≺ v n of P to disjoint chains C 1 , . . ., C t so that for each i, the chain C j to which v i is assigned, is determined solely by the subposet P v i .This formalizes the scenario in which the algorithm A receives the vertices of P one at a time, and when a vertex is received, irrevocably assigns it to one of the chains.Let χ A (P ≺ ) denote the number of (nonempty) chains that A uses to partition P ≺ , and χ A (P ) = max ≺ (χ(P ≺ )) over all presentation orders ≺ for P .For a class of posets P, let val A (P) = max P ∈P (χ A (P )) and val(P) = min A (val A (P)) over all online chain partitioning algorithms A. Our goal is to bound val(P w ), where P w is the class of finite posets of width w (allowing countably infinite posets with w finite in P w would not effect results).By Dilworth's Theorem [8], every poset with finite width w can be partitioned into w chains, and this is best possible.However this bound cannot be achieved online.In 1981, Kierstead proved Theorem 1 ( [15]).4w − 3 val(P w ) 5 w −1 4 .Kierstead asked whether val(P w ) is polynomial in w, and noted that his methods also provided a super linear lower bound.Until recently, there was little progress.Szemerédi (see [16]) proved a quadratic lower bound, which was improved to (2 − o(1)) w+1 Theorem 3. val(P w ) w 6.5 lg w+7 .
The difference between the proof of Theorem 1 and the proofs of Theorems 2 and 3 is fundamental.In the former relations are added to the online poset P ≺ to create a new online poset Q ≺ with smaller width so that every online chain of Q can be partitioned into 5 online chains of P ; then induction is applied.In the latter relations are deleted from P ≺ to form an online poset Q ≺ with the same width; this would seem to make it harder to partition Q, but paradoxically limits the wrong choices an algorithm can make.
The simplest online chain partitioning algorithm is First-Fit, which assigns each new vertex v i to the chain C j , with the least index j ∈ Z + such that for all h < i if v h ∈ C j then v h is comparable to v i .It was observed in [15] that val FF (P w ) = ∞ (see [16] for details) for any w > 1.The poset used to show this fact contains substructures that are important to this paper, so we present it.Lemma 4 ([15]).For every n ∈ Z + there is an online poset R ≺ n with width(R ≺ n ) 2 and χ FF (R ≺ n ) = n.
Proof.We define the online poset R ≺ n = (X, R , ≺) as follows.The poset R n consists of n chains X 1 , . . ., X n with and the additional comparabilities and incomparabilities given by: Note that the superscript of a vertex indicates to which chain X k it belongs and the subscript is its index within that chain.The example of R 5 is illustrated in Figure 1.The presentation order ≺ is given by X 1 ≺ • • • ≺ X n , where the order ≺ on the vertices of X k is the same as R on X k .
Observe that X k−2 R X k .Hence, the width of R n is 2. By induction on k one can show that each vertex x k i is assigned to chain C i .Despite Lemma 4, the analysis of the performance of First-Fit on restricted classes of posets has been useful and interesting.For posets P and Q, we say P is Q-free if P does not contain Q as an induced subposet.Let Forb(Q) denote the family of Q-free posets, and Forb w (Q) denote the family of Q-free posets of width at most w.Abusing notation, we write val FF (Q, w) for val FF (Forb w (Q)).
, and x i L y j for 1 j < i m.The vertices x 1 , . . ., x m are the lower leg and the vertices y 1 , . . ., y m are the upper leg of L m .The vertices x i , y i together form the i-th rung of L m .We provide a Hasse diagram of L m in Figure 1.Notice that for two consecutive chains X i and X i+1 of R n , the set Our attack is based on the following observation of Bosek and Krawczyk, first mentioned in [3,4], but never proved so far.Lemma 6. val(P w ) w val FF (L 2w 2 +1 , w) for w ∈ Z + .
In this paper we provide the first proof of the above-mention lemma.Kierstead and Smith completed this attack with the next lemma.Lemma 7 ([22]).val FF (L m , w) w 2.5 lg(2w)+2 lg m for m, w ∈ Z + .
Combining Lemmas 6 and 7 we get val(P w ) w 6.5 lg w+7 , which completes the proof of Theorem 3. Beside that, the paper presents two new constructions to show that the bounds given in Lemmas 6 and 7 can not be improved substantially and hence a new technique will be needed to prove a polynomial upper bound on val(P w ).
This paper is organized as follows.Section 2 introduces some notation and definitions.In Section 3 we present our online algorithm and reduce the proof of its performance bound to proving Lemmas 6 and 7, which are shown in Sections 4 and 5.In Section 6 we present constructions that show limitations of our approach.Section 7 contains some concluding observations.

Preliminaries
Let P = (V, P ) be a poset with u, v ∈ V .We usually write u ∈ P for u ∈ V (P ).The upset of u in P is U P (u) = {v : u < P v}, the downset of u in P is D P (u) = {v : v < P u}, and the incomparability set of u in P is I P (u) = {v : v P u}.The closed upset and closed downset of u in P are, respectively, U P [u] = U P (u) + u and The subposet of P induced by U is denoted by P [U ], and be the set of maximal vertices in P [U ] and Min P (U ) be the set of minimal vertices in P [U ].Let Max P = Max P (V ) and Min P = Min P (V ).
A chain partition C of P is a Dilworth partition if |C| = width(P ).If vertices u and v are in the same chain of some Dilworth partition then uv is called a Dilworth edge of P .
Let M P = (V P , ⊑ P ), where V P is the set of maximum antichains in P and ⊑ P is defined by If A ⊑ P B and A = B, we write A ❁ P B. In [9] Dilworth showed that M P is a lattice with the meet and the join defined by A poset P = (V, P ) is bipartite if the set V can be partitioned into two disjoint antichains A, B such that A ❁ P B -such a poset is denoted by (A, B, P ).A bipartite poset P = (A, B, P ) is a core if |A| = |B| and for any comparable pair x P y with x ∈ A and y ∈ B, xy is a Dilworth edge (see Figure 2).Informally, we think of a core as a bipartite poset whose Hasse diagram is a balanced bipartite graph in which each edge is included in some perfect matching.
Poset Q is a core of width 5. R is not a core since xy is not a Dilworth edge.
A chain in a poset P corresponds to an independent set in its cocomparability graph.Offline the terms chain partition and coloring are interchangeable, but an online chain partitioning algorithm has more information to use than an online coloring algorithm.This advantage is lost by First-Fit.
The notion of Grundy coloring is useful for analyzing First-Fit.
, there is some u ∈ P so that g(u) = i (i.e.: g is surjective); and (G3) if v ∈ P with g(v) = j, then for all i ∈ [j − 1] there is some u ∈ I P (v) such that g(u) = i.
Often, we call the elements of [n] as colors.If u ∈ P and g(u) = i, we say u is colored with i.Let u, v ∈ P .If u P v and g(u) < g(v), we say u is a g(u)-witness for v under g.
The next lemma is folklore.It allows the analysis of a dynamic online process in a static setting.
Lemma 9.For any poset P , the largest n for which P has an n-Grundy coloring is equal to χ FF (P ).

The online algorithm
In this section we provide a simple online algorithm A for chain partitioning online posets.In the next two sections we show that A achieves the performance bound stated in Theorem 3. If W is a subset of P , we set W x = W ∩ {y : y x} and W ≺x = W ∩ {y : y ≺ x}.
(1) Construct an online partition V = X 1 ∪ • • • •∪ • X width(P ) by putting every consecutive vertex x of (V, ≺) to the set X w , where w is the least integer with width(P (2) For every w ∈ [width(P )], construct an on-line poset R ≪ w , where R w = (Z, R ), together with an injection φ : X w → Z that satisfies the property that R w [φ(X w )] is a subposet of P [X w ].Thus, a partition of R ≪ w into chains yields a partition of P [X w ] into chains.This more complex procedure is explained in Subsection 3.2.
(3) For every w ∈ [width(P )], use First-Fit to partition R ≪ w into chains.The final chain partition consists of all chains produced by Procedure (3) for w = 1, . . ., width(P ). 1n Section 4 we show that R w is a (2w2 + 1)-ladder free poset of width w.In Section 5 we show that val F F (L m , w) w 2.5 lg(2w)+2 lg m .
Then, since a chain partition of R ≪ w yields a chain partition of P [X w ] with at most the same number of chains, Theorem 3 follows by (1) val(P w ) , w) w 2.5 lg(2w)+2 lg(2w 2 +1)+1 w 6.5 lg w+7 .
In the remaining of the paper, we write R ≪ and R instead of R ≪ w and R w whenever w is clear from the context.
x and the ⊑ P -chain and The maximum antichains A x in A are computed in the order in which the elements x are added to the set X w .So, we may view ( A, P ) as an on-line poset with the presentation order extended by the elements from A x \ {A y : y ∈ X ≺x w } each time a new antichain A x from A is computed.It is likely that the antichains in A are not disjoint.In the next step we slightly modify this poset by making these antichains pairwise disjoint.
x , D x , and U x .
Define the presentation order ≪ of the poset U = (Z, U ) by putting B x ≪ B y if x ≺ y for x, y ∈ X w , and by arbitrarily ordering the elements in B x for x ∈ X w .
By (3), B x is an w-antichain in U .Indeed, if S is a (w + 1)-subset in Z then there are distinct y, z ∈ S with comparable (possibly identical) first coordinates.By (3), they are comparable in U .Thus width(U ) = w, and B partitions Z into disjoint maximum antichains.Moreover, B is a ⊑ U -chain.Note that the antichain B x and the relation U between the elements in B x and the elements in the set Z ≺x can be computed when x is processed.
In its last step, Procedure (2) constructs an online poset R ≪ = (Z, R , ≪), where R = (Z, R ) is obtained from (Z, U ) by deleting some non-Dilworth edges of (Z, U ). Suppose R restricted to Z ≺x is already computed.When x is processed, the edges (v, u) of We prove this by induction on ≺.The claim holds for the set Z y , where y is the first vertex in (V, ≺) added to X w .Suppose the claim holds for Z ≺x .Assume that v ≪ u and u ∈ B x (the other cases are handled similarly).Then there is a Dilworth partition C with a chain a chain partition of P [X w ] into at most the same number of chains: indeed, it is enough to assign x ∈ X w to a chain labeled i if φ(x) = (x, A x ) is assigned to a chain i.
Lemma 10.The relation R = (Z, R ) is a width w poset, and for all x, y ∈ X w with Proof.Conditions (R1) and (R2) follow immediately from the definition of R.
First we prove that R is a poset of width w.As R is obtained from the poset U by removing some non-loops, R is reflexive and antisymmetric.For transitivity, argue by induction on ≺.Suppose u < R v < R w.Then there are distinct x, y, z ∈ X w with u ∈ B x , v ∈ B y , w ∈ B z , and We prove (R6) by induction on ≺.Suppose u ∈ B x and v ∈ B y with u R v. The claim holds if z = x or z = y.Suppose z ∈ X w is such that we are done.Otherwise, as z ≺ p(y), there is w ∈ B z with u < R w < R w ′ < R y by induction, and hence (R6) holds.Suppose y ≺ x.By (R2), there is In [3,4] a width w poset R is defined to be regular if it has a partition B satisfying (R1)-(R4).An example of a regular poset is given in Figure 5.
In this section we prove that R ∈ Forb w (L 2w 2 +1 ), which yields Lemma 6 as a corollary.For any u ∈ R, let B(u) be the antichain with u ∈ B(u) ∈ B. Consider an arbitrary m-ladder Proof.See Figure 6.As width(R) w, it suffices to show by induction on i that The base step i = 1 holds, since ] is a core with Dilworth edge zy i .Let C be a Dilworth partition of R[B(y i−1 ) ∪ B(y i )] with z and Figure 5. Regular poset P ≪ with P = ( 6i=1 A i , P ) and the presentation order given by A 1 ≪ A 2 ≪ . . .≪ A 6 (the order ≪ inside each A i is arbitrary).y i in the same chain.Each vertex of S is matched in C to a distinct vertex of B(y i ), different than y i (see Figure 6) as z / ∈ S. Proof.See Figure 7. Assume to the contrary that B(x 2w+1 ) ❁ R B(y 1 ).It follows that ; else set I = {w + 1, . . ., 2w + 1}.Regardless, |I| = w + 1, and by (R6), there are z i with x i < R z i < R y i for all i ∈ I (see Figure 7).As |B z | = w there are i, j ∈ I with i < j and z i = z j .Then x j < R z j = z i < R y i , a contradiction with x j R y i .

First-Fit on ladder-free posets
In this section we prove val FF (L m , w) w 2.5 lg(2w)+2 lg m for m, w ∈ Z + , which shows Lemma 7.This proof was already published in [22], here we present its shortened version to keep the paper self-contained.Consider a Grundy coloring g of a poset P .Let C = {x 1 , . . ., x k } be a chain of P such that g(x 1 ) < • • • < g(x k ); we call C ascending if x 1 < P x 2 < P . . .< P x k (see Figure 8) and we call C descending if x 1 > P x 2 > P . . .> P x k .
The next two propositions are the combinatorial tools for the upcoming arguments in the proof of Lemma 7. The first one is just a restatement of the Erdős-Szekeres Theorem and the second one presents conditions for ascending and descending chains in a poset with forbidden ladder.Proposition 14.Consider a poset P and its Grundy coloring g.Let C be a chain in P such that all g(c) for c ∈ C are distinct.If the length of every ascending subchain of C is at most s and the length of every descending subchain C is at most t, then |C| st.
Proposition 15.Suppose P ∈ Forb(L m , w) and w 2. Let x 1 < . . .< x k be an ascending (resp.let x 1 > . . .> x k be a descending) chain in P and for each i ∈ [k − 1] let y i be a g(x i )-witness for x i+1 .Then for all i, j with 1 i < j k, (C1) x i < P y i (resp.x i > P y i ); (C2) y i > P x j (resp.y i < P x j ); and (C3) if Proof.We consider the case that x 1 < . . .< x k is an ascending chain; the other case can be proved analogically.Observe that (C1) and (C2) follow immediately from definitions (see Figure 8).To show (C3) assume to the contrary that k > m(w − 1).Then k (m − 1)(w − 1) + 2 as w 2. The subposet P 0 := P [{y 1 , . . ., y k−1 }] has width at most w − 1 as y h P x k for all h ∈ [k − 1] by hypothesis.Since |P 0 | (m − 1)(w − 1) + 1, there is a chain y i 1 < P . . .< P y im in P 0 , by Dilworth's Theorem.By (C2), i 1 < . . .< i m .Thus by (C1) and hypothesis, P [{x i 1 , y i 1 , . . ., x im , y im }] is an m-ladder, contradicting P ∈ Forb(L m , w).P g(x 1 ) P g(x 2 ) P g(x 3 ) P g(x 4 ) P g(x 5 ) P g(x 6 ) x 1 x 2 x 3 x 4 x 5 x 6 Proof of Lemma 7. We argue by induction on w = width(P ).The base step w = 1 is trivial.Now fix w, and assume the lemma holds for all smaller values of w.
Let P = (V, P ) be a poset of width w such that P ∈ Forb(L m ), let g : V → [n] be an n-Grundy coloring of P with n = val FF (L m , w), and let C i = g −1 (i).We must show that n w 2.5 lg(2w)+2 lg m .
Pick a maximum antichain A ∈ V P with N := min a∈A g(a) maximum, i.e., .
Let D = D(A) and U = U (A).As A is a maximum antichain, , set q − i = max(C i ∩ D) and q + i = min(C i ∩ U ). Hence q − i and q + i are consecutive on C i .Each a ∈ A has an i-witness, and so satisfies q − i P a or q + i P a. Thus, q − i or q + i is special.Pick a special vertex q i ∈ {q − i , q + i } and pick r i ∈ C i so that r i is a minimal special vertex in C i if q i ∈ D, and r i is a maximal special vertex in C i if q i ∈ U (it might happen that r i = q i ).Call q i the near witness and r i the far witness.Set R = {r 1 , . . ., r N −1 }.The next claim completes our recursion for val F F (L m , w).
Recall that all g(r i ) are distinct.It gives that also all g(s), for s ∈ S, are distinct.Therefore, by Proposition 14, it suffices to show: (T1) the size of any ascending chain in S is at most m(w − 1); and (T2) the size of any descending chain in S is at most w 2 m(w − 1) val FF (L m , ⌊w/2⌋).For (T1), let x 1 < P . . .< P x k be any ascending chain in S. For each i ∈ [k − 1], pick a g(x i )witness y i for x i+1 .Using Proposition 15, (C2) implies y i > P x k .Suppose y i < P x k .By (C1), x i < P y i .Then y i ∈ U and y i is special as x k is special.As g(x i ) = g(y i ) this contradicts the choice of x i as a far witness.Thus y i P x k .By (C3), |S| = k m(w − 1).
For i ∈ [k], let w i be the near witness for color g(z i ).Note that w i ∈ P ′ since w i z i z 1 .By Dilworth's Theorem, there is a chain T ⊆ {w 1 , . . ., w k } with k w 2 |T |.Each w i has different Groundy color, thus by Proposition 14, it suffices to prove: (T3) the size of any descending sequence in T is at most m(w − 1), (T4) the size of any ascending sequence in T is at most val FF (L m , ⌊ w 2 ⌋).For (T3), let s 1 > P . . .> P s l be a descending chain in T .For 1 i l − 1, pick a g(s i )-witness t i of s i+1 .Using Proposition 15, (C2) implies t i < P s l .Suppose s l < P t i .Then t i ∈ U and t i is also special as t i < P s i by (C1).As g(t i ) = g(s i ) this contradicts the choice of s i as a near witness.Thus t i P s l .By (C3), |T | = l m(w − 1).
For (T4), let u 1 < P . . .< P u l be an ascending chain in T , and for i ∈ [l] let v i ∈ S ′ be the far g(u i )-witness.Then u l P v l .As S ′ is descending, Then g ′ is an l-Grundy coloring of P ′′ : as g is a Grundy coloring, if i < j and y ∈ U j then there is x ∈ C i with x P y; as u i < P u j P y P v j < P u i , we have x ∈ U i .Since Consider a Dilworth chain decompositions of R and let S be a chain with a maximum size in this decomposition.Since the width of R is at most w, we have After applying Claim 15.A we get The equation ( 5) with n = val FF (L m , w) can be now rewrite into the following recursion Applying this recursion repeatedly to the second term, with val FF (L m , 1) = 1, we obtain

Arguing by induction yields:
val FF (L m , w) m 2 lg w w 2.5 lg(2w) w 2.5 lg(2w)+2 lg m , which completes the proof of the lemma.

Limitations of Our Methods
Loosely speaking, two major parts of the proof of our main theorem rely on limiting the number of rungs in a ladder within a regular poset and the performance of First-Fit on the family Forb(L m ).Here, we will show that our general upper bound for the online coloring problem cannot be greatly improved with our current methods.
In the first part of the section we show that the assertion of Lemma 13 can not be improved.Although L 2w 2 +1 is not a subposet of any width w regular poset, we show that there are regular posets of width w that contain ladders whose number of rungs is quadratic in w.
Lemma 16.For each integer w 2, there is a regular poset P ≪ so that width(P ) = w and P contains L w⌊(w+2)/2⌋ as an induced subposet.
Proof.Consider two antichains A = {u 1 , . . ., u w } and B = {v 1 , . . ., v w }, where u i and v i are the i-th elements of A and B, respectively.We say (A, B, ) is a core of: It is straightforward to verify that bipartite posets of types I, S k and T k are cores.See Figure 9 for examples.
Hasse diagrams of I, S 6 , and T 4 for w = 6.Now we construct an auxiliary regular poset Q ≪ , based on which the regular poset P ≪ is built.Let V = 2w+1 i=1 A i , and let the presentation order of Q ≪ be A 1 ≪ A 2 ≪ . . .≪ A 2w+1 .The poset Q = (V, Q ) is defined as follows.At every step in the presentation of Q, every two ⊑ Q -consecutive antichains induce a core, one of type: I, S k or T k for k ∈ [w].Suppose that the antichains A s(i) and A p(i) , if exist, denote the antichains that are respectively just above and just below A i at the moment A i is presented.To define the relation Q in Q we need only to determine the relation Q between A i and A s(i) and between A p(i) and A i at the moment A i is presented; the other comparabilities will follow by transitivity -see (R2).Below are the rules how Q is determined for the successively presented antichains A 1 , . . ., A 2w+1 : (1) A 2 is set so that The above rules imply the following relations between the antichains A 1 , . . ., A 2w+1 in the poset Q (see Figure 10): In the last part of this section we give the lower bound on val FF (L m , w), which shows that the upper bound from Lemma 7 can not be substantially improved.For the upcoming construction we remind the definition of the lexicographical product of two posets.For posets P and Q, the lexicographical product P • Q is the poset with vertices {(p, q) : p ∈ P, q ∈ Q} and order P •Q , where (p 1 , q 1 ) P •Q (p 2 , q 2 ) if either p 1 < P p 2 or (p 1 = p 2 and q 1 Q q 2 ).Informally, we may think of P • Q as the poset P where each vertex has been "inflated" to a copy of Q.It is well know that (10) width(P • Q) = width(P ) width(Q).
The following two simple properties (we left the proof for the reader) are the key in the proof of the upcoming lemma.For p, r ∈ P and u, v, s ∈ Q we have: Lemma 17.For m, w ∈ Z + with m > 1, we have w lg(m−1) /(m − 1) val FF (L m , w).
Proof.Fix m ∈ Z + with m > 1.Let R be the width 2 poset R m−1 as defined in the proof of Lemma 4. For technical reasons we would like R to have the least and the greatest element.Vertex x m−1 1 is already the greatest in R, but there is no least element in R. Therefore we extend R to P by adding a new element 0 which is below entire R. The greatest element in P is still x m−1 1 , which we denote by 1.
It is a simple exercise to see that P also satisfies the statement of Lemma 4, i.e., width(P ) = 2 and χ FF (P ) χ FF (R) m − 1.As R is an induced subposet of P we have I P ( 0) = ∅ and Observe that in a ladder L m , the lowest vertex of the upper leg is always incomparable to m − 1 vertices.Hence, there is no vertex in P that can serve as the lowest vertex of the upper leg of an m-ladder and thus (13) P ∈ Forb(L m ).
We are prepared to build a poset Q k ∈ Forb(L m ) with n-Grundy coloring so that width(Q k ) = 2 k and n (m − 1) k .Poset Q k is defined by the following rules: Note that Q 1 and P are isomorphic and so we will treat Q 1 as P .The next two properties are the consequence of the definition of Q k , equation (10) and the fact that P has the least and the greatest element with width(P ) = 2.For each k ∈ N (Q3) Q k has a minimum vertex and a maximum vertex, (Q4) width( Proof.We will use induction on k.For our bases, we see k = 0 is trivial and k = 1 is established by (13).Take k > 1 and suppose the inductive hypothesis holds for all smaller cases.Assume L is an m-ladder in Q k with the lower leg (a 1 , u If all vertices of L are pairwise different in the first coordinate, then these vertices would induce an m-ladder in P , which violates (13).Hence, at least two vertices of L share a first coordinate, say p ∈ P .Let Q ′ = {(p, q) : q ∈ Q k−1 } and note that Q ′ and Q k−1 are isomorphic.Let 0 and 1 to be the minimum and the maximum, respectively, vertices of Q ′ (which exist by (Q3)).
Assume for a while, Q ′ contains two vertices of the lower leg of L, i.e., there are i < j ∈ [m] so that (a i , u i ), (a j , u j ) ∈ Q ′ with a i = a j = p.From the definition of a ladder, we know (a i , u (11) we have b i = p and thus Q ′ contains (b i , v i ), a vertex of the upper leg of L. For similar reasons, if Q ′ contains two vertices of the upper leg of L, then it has to have one of the lower leg of L. Therefore, there are (a i , u i ), (b j , v j ) ∈ Q ′ , vertices of the lower and the upper leg of L, respectively.We see (a i , u In the former case we use (11) and in the latter case (12) ).Again, using (11) or (12), we have (b 1 , v 1 ) ∈ Q ′ .
It is easy to check that a function (f, g) → (f − 1)n + g is a bijection between [m − 1] × [n] and [(m − 1)n].Since f and g are surjective, then also h must be surjective.Thus, h satisfies (G2).To show (G1) suppose h((p, q)) = h((r, s)).This implies that f(p) = f(r) and g(q) = g(s).By (G1) of f and g, two pairs of vertices p, r and q, s are comparable respectively in P and in Q k .Therefore, by the definition of the lexicographical product, vertices (p, q) and (r, s) are comparable in Q k+1 and condition (G1) holds for h.
Consider (r, s) ∈ Q k+1 so that h((r, s)) = j > 1 and take any i < j.We will show (r, s) has an i-witness in Q k+1 which will prove (G3).There are unique integers c ∈ [m − 1] and d ∈ [n] so that j = (c − 1)n + d and f(r) = c, g(s) = d.Similarly, we can find a ∈ [m − 1] and b ∈ [k] so that i = (a − 1)n + b.As i < j, we must have a c.
Suppose a = c, then b < d.As g satisfies (G3), there is some q ∈ Q k so that g(q) = b and q Q k s.By the definition of lexicographical product, (r, q) Q k+1 (r, s).Observe h((r, q)) = i and then (r, q) is the desired witness.
The case a < c is similar.This time we use (G3) of f to get p ∈ P so that f(p) = a and p P r.Take any q ∈ Q k so that g(q) = b (q exists by (G2) of g).Again, by the definition of lexicographical product, (p, q) Q k+1 (r, s).Finally, as h((p, q)) = i, we deduce (p, q) is the desired witness in this case.
Lemmas 16 and 17 show that the upper bound of val(P w ) cannot be pushed below w lg w using our current methods.

Concluding Remarks
Although we have improved the upper bound for val(P w ), our current methods cannot bring it down to a polynomial bound without some major changes.Perhaps improvements in the understanding of regular posets could lead us to a subfamily of more interesting forbidden substructures.We could also examine online coloring algorithms other than First-Fit to reduce the number of colors used on the family Forb(L m ).
We may look beyond the scope of val(P w ).So far, the reduction to regular posets has only been studied on general posets.We might ask what the results of Procedures (1) and ( 2) are when we start with a poset from Forb(Q) (for some poset Q).It is interesting to ask what analogues of the inequality (1) could be built.For instance, could an analogue for cocomparability graphs be created?Already, Kierstead, Penrice, and Trotter [19] have shown that a cocomparability graph can be colored online using a bounded number of colors.However, this bound is so large that it was not computed.Perhaps methods similar to the reduction to regular posets could be created.
and u < R w by induction.The other two cases are similar, but easier.Thus R is a poset.As no Dilworth edges are removed from U to form R, width(R) = width(U ) = w.Thus (R3) and (R4) also hold.We prove (R5) by induction on ≺.Assume x ≺ y.The case y ≺ x is dual.By (R1), R[B p(y) ∪B y ] is a core.If x = p(y) we are done.Otherwise, R[B x ∪ B p(y) ] is a core by induction.Thus R[B x , B y ] is a core by definition of R .So, (R5) holds.

5 Figure 8 .
Figure 8.The ascending chain x 1 < P . . .< P x 6 .The point y i is a g(x i )-witness of x i+1 .

Figure 10 .
Figure 10.The width 5 poset Q is shown on the right.The sketch of the construction of the width 5 poset P is shown on the left.It consists of 3 copies of Q (the middle copy of Q in P is depicted with gray background) joined as shown in the figure.