Cyclic Van der Waerden Numbers

• Cyclic Van der Waerden numbers are defined as the smallest modulus N such that every r-coloring of Z_M (for all M ≥ N) contains a monochromatic k-term cyclic arithmetic progression.
• The theory employs hypergraph formulations and layered constructions to derive explicit lower bounds for independence and chromatic numbers in cyclic groups.
• It bridges classical Van der Waerden numbers with cyclic variants by harnessing modular arithmetic and explicit combinatorial constructions to improve known thresholds.

Cyclic Van der Waerden numbers are Ramsey-type thresholds for monochromatic arithmetic progressions in finite cyclic groups. For integers $k\ge 3$ and $r\ge 2$, the cyclic Van der Waerden number $W_c(k,r)$ is the least modulus $N$ such that for every $M\ge N$, every $r$-coloring of $\mathbb{Z}_M$ contains a monochromatic $k$-term cyclic arithmetic progression modulo $M$. This cyclic formulation was introduced by Burkert and Johnson as a mechanism for obtaining lower bounds on the classical Van der Waerden numbers $W(k,r)$, and recent work has developed a hypergraph-theoretic and constructive theory that yields explicit lower bounds for $r\ge 2$0, especially when the modulus is a multiple of the progression length (Liber, 9 Sep 2025).

1. Definitions and basic framework

For $r\ge 2$1, write $r\ge 2$2. A $r\ge 2$3-term cyclic arithmetic progression modulo $r\ge 2$4, with $r\ge 2$5, is a $r\ge 2$6-element subset of $r\ge 2$7 of the form

$r\ge 2$8

for some base $r\ge 2$9 and common difference $W_c(k,r)$0. The tuple notation emphasizes the progression structure, but combinatorially the object is a set rather than an ordered sequence. Different choices of $W_c(k,r)$1 and $W_c(k,r)$2 may generate the same subset. If $W_c(k,r)$3 is valid, then so is $W_c(k,r)$4, so one may always choose a representative with $W_c(k,r)$5; the paper calls the smallest such $W_c(k,r)$6 the common difference of the progression (Liber, 9 Sep 2025).

A subset $W_c(k,r)$7 is $W_c(k,r)$8-term cyclic-AP-free, or $W_c(k,r)$9-AP-free mod $N$0, if it contains no $N$1-term cyclic arithmetic progression modulo $N$2. This notion is the cyclic analogue of progression-free subsets of intervals in the classical setting.

The classical Van der Waerden number $N$3 is the least $N$4 such that every $N$5-coloring of $N$6 contains a monochromatic ordinary $N$7-term arithmetic progression. By contrast, $N$8 is the least modulus $N$9 such that for all $M\ge N$0, $M\ge N$1 admits no $M\ge N$2-coloring avoiding monochromatic $M\ge N$3-term cyclic arithmetic progressions. Equivalently, $M\ge N$4 is the smallest $M\ge N$5 such that for every $M\ge N$6, every $M\ge N$7-coloring of $M\ge N$8 contains a monochromatic cyclic $M\ge N$9-AP (Liber, 9 Sep 2025).

2. Hypergraph formulation, independence, and coloring

The standard combinatorial encoding is the cyclic Van der Waerden hypergraph

$r$0

where $r$1 and $r$2 is the set of all $r$3-term cyclic arithmetic progressions modulo $r$4. A subset $r$5 is independent in $r$6 exactly when it is $r$7-AP-free mod $r$8. The quantity

$r$9

is therefore the independence number of $\mathbb{Z}_M$0. In the case $\mathbb{Z}_M$1, this becomes $\mathbb{Z}_M$2, the largest size of a cyclic $\mathbb{Z}_M$3-AP-free subset of $\mathbb{Z}_M$4 (Liber, 9 Sep 2025).

The corresponding coloring parameter is

$\mathbb{Z}_M$5

defined as the minimum number of subsets needed to partition $\mathbb{Z}_M$6 into parts each of which is $\mathbb{Z}_M$7-AP-free mod $\mathbb{Z}_M$8. Equivalently, $\mathbb{Z}_M$9 is the chromatic number of $k$0. Since a proper $k$1-coloring partitions the vertex set into $k$2 independent sets,

$k$3

The bridge to cyclic Van der Waerden numbers is immediate. If $k$4, then $k$5 has an $k$6-coloring with no monochromatic $k$7-term cyclic AP, so

$k$8

Accordingly, lower bounds on independence numbers and explicit proper colorings of $k$9 translate directly into lower bounds on $M$0 (Liber, 9 Sep 2025).

3. Arithmetic structure of cyclic progressions modulo $M$1

A central structural invariant is

$M$2

The key constraint is the cyclic length condition: if there exists a $M$3-term cyclic AP modulo $M$4 with common difference $M$5, then

$M$6

Conversely, if $M$7 and $M$8, then such a progression exists. Specializing to $M$9 yields an explicit characterization: $W(k,r)$0 Thus the relevant $W(k,r)$1-values are precisely the divisors of $W(k,r)$2 not exceeding $W(k,r)$3 (Liber, 9 Sep 2025).

This arithmetic description sharply restricts the progression types that must be controlled. It also induces an increasing sequence

$W(k,r)$4

of allowed divisor-values, and this ordered set drives Liber’s layered construction of large cyclic-AP-free subsets.

Berglund proved the general upper bound

$W(k,r)$5

Moreover, if $W(k,r)$6, then equality holds: $W(k,r)$7 Liber shows that whenever $W(k,r)$8, the upper bound is strict: $W(k,r)$9 The argument is extremal: if a $r\ge 2$00-AP-free set had size $r\ge 2$01, then its complement would be exactly $r\ge 2$02; the existence of a second allowable difference $r\ge 2$03 would then force a $r\ge 2$04-term cyclic AP entirely inside the large set, a contradiction (Liber, 9 Sep 2025).

Two further structural lemmas underlie the construction. First, if $r\ge 2$05 and $r\ge 2$06 is a cyclic AP mod $r\ge 2$07 with common difference $r\ge 2$08, then all elements of $r\ge 2$09 lie in a single congruence class modulo $r\ge 2$10. Second, if one arranges the elements of $r\ge 2$11 as an $r\ge 2$12 array, then any $r\ge 2$13-term AP with allowable common difference $r\ge 2$14 must pass through a block of $r\ge 2$15 consecutive positions in an appropriate vertical slice. This “vertical block” phenomenon is what the construction exploits (Liber, 9 Sep 2025).

4. Explicit construction of large cyclic-AP-free sets

Liber constructs a set $r\ge 2$16 such that its complement

$r\ge 2$17

is $r\ge 2$18-AP-free mod $r\ge 2$19. Writing

$r\ge 2$20

and setting $r\ge 2$21, define for each $r\ge 2$22

$r\ge 2$23

and then

$r\ge 2$24

Each $r\ge 2$25 is a union of tail segments from vertical arithmetic progressions

$r\ge 2$26

A crucial fact is that distinct layers do not overlap: $r\ge 2$27 Hence

$r\ge 2$28

and direct counting gives

$r\ge 2$29

Therefore

$r\ge 2$30

The proof that $r\ge 2$31 is $r\ge 2$32-AP-free combines the congruence-class containment lemma with a covering lemma for cyclic intervals. Fix a $r\ge 2$33-term cyclic AP $r\ge 2$34 with common difference $r\ge 2$35. Because $r\ge 2$36 lies in one congruence class modulo $r\ge 2$37, the vertical-slice lemma implies that $r\ge 2$38 contains a block of $r\ge 2$39 consecutive elements in an appropriate slice

$r\ge 2$40

A simple covering argument shows that every cyclic interval of length $r\ge 2$41 in $r\ge 2$42 meets the tail $r\ge 2$43; transported back to $r\ge 2$44, that tail is exactly one of the pieces placed into $r\ge 2$45. Hence every $r\ge 2$46-term cyclic AP meets $r\ge 2$47, so none is contained in $r\ge 2$48 (Liber, 9 Sep 2025).

The resulting theorem is

$r\ge 2$49

This is an explicit, computable lower bound for the independence number of $r\ge 2$50, valid for all $r\ge 2$51 with $r\ge 2$52 (Liber, 9 Sep 2025).

5. Chromatic numbers and explicit lower bounds for $r\ge 2$53

The same construction can be refined from a single large independent set to full partitions of $r\ge 2$54 into a small number of $r\ge 2$55-AP-free parts. The results separate into three regimes (Liber, 9 Sep 2025).

Regime	Bound on $r\ge 2$56	Consequence for $r\ge 2$57
$r\ge 2$58	$r\ge 2$59	$r\ge 2$60 by taking $r\ge 2$61
$r\ge 2$62	$r\ge 2$63	$r\ge 2$64
$r\ge 2$65	$r\ge 2$66	$r\ge 2$67

When $r\ge 2$68, the partition $r\ge 2$69 already suffices: both $r\ge 2$70 and $r\ge 2$71 are $r\ge 2$72-AP-free, so $r\ge 2$73. The underlying reason is that the distinguished progression

$r\ge 2$74

has length $r\ge 2$75, so it is too short to be a forbidden $r\ge 2$76-term progression.

When $r\ge 2$77, the set $r\ge 2$78 itself contains a $r\ge 2$79-term progression,

$r\ge 2$80

so $r\ge 2$81 cannot be a single color class. Liber therefore splits $r\ge 2$82 into two sets $r\ge 2$83 and $r\ge 2$84 by alternating blocks of size $r\ge 2$85 in the increasing order of the elements of $r\ge 2$86. A case analysis shows that neither $r\ge 2$87 nor $r\ge 2$88 contains a forbidden $r\ge 2$89-term cyclic AP, whence $r\ge 2$90 can be partitioned into the three $r\ge 2$91-AP-free sets $r\ge 2$92.

When $r\ge 2$93, one first restricts to the initial block

$r\ge 2$94

which has the same pattern of allowable differences as in the case $r\ge 2$95, since

$r\ge 2$96

for $r\ge 2$97. The set $r\ge 2$98 is split into $r\ge 2$99 and $W_c(k,r)$00 exactly as before, while the remainder

$W_c(k,r)$01

satisfies $W_c(k,r)$02 and is partitioned into subsets $W_c(k,r)$03 of size at most $W_c(k,r)$04, where

$W_c(k,r)$05

Each $W_c(k,r)$06 is automatically $W_c(k,r)$07-AP-free by size, yielding the stated upper bound on $W_c(k,r)$08.

These coloring constructions produce a family of explicit lower bounds: $W_c(k,r)$09 and, for $W_c(k,r)$10,

$W_c(k,r)$11

6. Relation to earlier work and to classical Van der Waerden numbers

Burkert and Johnson introduced the cyclic Van der Waerden numbers $W_c(k,r)$12 as a tool for obtaining lower bounds on the classical Van der Waerden numbers $W_c(k,r)$13. Berglund then studied the independence numbers $W_c(k,r)$14, especially for $W_c(k,r)$15 with small $W_c(k,r)$16, establishing the general upper bound $W_c(k,r)$17, proving equality when $W_c(k,r)$18, and determining $W_c(k,r)$19 exactly via an odd-even dichotomy. Liber’s contribution is to extend Berglund’s methods to all $W_c(k,r)$20 through the divisor set $W_c(k,r)$21, the layered forbidden set $W_c(k,r)$22, and the resulting explicit lower bound for $W_c(k,r)$23 (Liber, 9 Sep 2025).

The cyclic formulation feeds back into the classical one through periodicity. In the discussion surrounding Burkert–Johnson’s program, any modulus $W_c(k,r)$24 for which there exists an $W_c(k,r)$25-coloring of $W_c(k,r)$26 avoiding cyclic $W_c(k,r)$27-APs gives a periodic coloring of $W_c(k,r)$28 avoiding ordinary $W_c(k,r)$29-APs on intervals of length at least $W_c(k,r)$30, implying a lower bound of the form

$W_c(k,r)$31

Consequently, the explicit cyclic bounds imply at least

$W_c(k,r)$32

with analogous implications for larger $W_c(k,r)$33 (Liber, 9 Sep 2025).

Conceptually, the cyclic theory isolates the arithmetic of allowable common differences in $W_c(k,r)$34. The set $W_c(k,r)$35, the subgroup structure encoded by $W_c(k,r)$36, and the $W_c(k,r)$37 array picture together make the cyclic setting particularly amenable to explicit constructions. This suggests that cyclic Van der Waerden numbers are not merely auxiliary bounds, but a distinct combinatorial theory whose invariants—independence numbers, chromatic numbers, and allowable-difference sets—admit more explicit control than is currently available in the classical interval setting.

7. Related cyclic and modular variants

Several adjacent theories illuminate the scope of the term “cyclic” in Van der Waerden-type problems. One is the zero-sum theory of arithmetic progressions. For $W_c(k,r)$38, the zero-sum number $W_c(k,r)$39 is the minimum integer such that every coloring $W_c(k,r)$40 admits a $W_c(k,r)$41-term arithmetic progression whose color-sum is $W_c(k,r)$42. That paper explicitly interprets these parameters as cyclic or modular versions of Van der Waerden’s theorem, but they differ from $W_c(k,r)$43: the ambient set is the interval $W_c(k,r)$44, not $W_c(k,r)$45, and the forced pattern is zero-sum rather than monochromatic (Robertson, 2018).

A second adjacent notion is the anti-Van der Waerden theory on cycles. For graph colorings, the anti-Van der Waerden number $W_c(k,r)$46 is the least number of colors forcing a rainbow $W_c(k,r)$47-term arithmetic progression. In the case of cycles, the set of arithmetic progressions on $W_c(k,r)$48 is isomorphic to the set of arithmetic progressions on $W_c(k,r)$49, so $W_c(k,r)$50. This again concerns cyclic ambient structure, but it studies rainbow rather than monochromatic phenomena (Miller et al., 2022).

Topological encodings of arithmetic progressions form a third nearby direction. The van der Waerden complex $W_c(k,r)$51 is the simplicial complex on $W_c(k,r)$52 whose facets correspond to arithmetic progressions of length $W_c(k,r)$53, and it is homotopy equivalent to a CW-complex whose cells asymptotically have dimension at most $W_c(k,r)$54. That work is entirely linear rather than cyclic. A plausible implication is that a cyclic analogue of the van der Waerden complex could provide a topological model for the hypergraphs $W_c(k,r)$55, but such a construction is not developed there (Ehrenborg et al., 2016).

Taken together, these variants delimit the specific meaning of cyclic Van der Waerden numbers in the sense of Burkert and Johnson. The defining features are the ambient cyclic group $W_c(k,r)$56, monochromatic $W_c(k,r)$57-term cyclic arithmetic progressions, and the eventual-threshold quantifier “for every $W_c(k,r)$58.” Within that framework, the current theory is driven by explicit constructions of large independent sets in cyclic Van der Waerden hypergraphs and by the conversion of those constructions into lower bounds for $W_c(k,r)$59 (Liber, 9 Sep 2025).