European Journal of Mathematics

, Volume 1, Issue 2, pp 320–328 | Cite as

Abelian groups yield many large families for the diamond problem

  • Éva Czabarka
  • Aaron Dutle
  • Travis Johnston
  • László A. Székely
Research Article

Abstract

There is much recent interest in excluded subposets. Given a fixed poset \(P\), how many subsets of \([n]\) can be found without a copy of \(P\) realized by the subset relation? The hardest and most intensely investigated problem of this kind is when \(P\) is a diamond, i.e. the power set of a two-element set. In this paper, we show infinitely many asymptotically tight constructions using random set families defined from posets based on Abelian groups. They are provided by the convergence of Markov chains on groups. Such constructions suggest that the diamond problem is hard.

Keywords

Poset Diamond-free families Lubell function Boolean lattice Abelian group Markov chain 

Mathematics Subject Classification

05D05 06A06 11B13 11P70 20K01 

1 Introduction

This introduction largely follows the concise and accurate description of the background and history from [20]. For posets \(P = (P,\le )\) and \(P' = (P',\le ')\), we say \(P'\) is a weak subposet of \(P\) if there exists an injection \(f :P' \rightarrow P\) that preserves the partial ordering, meaning that whenever \(u \le ' v\) in \(P'\) we have \(f(u) \le f(v)\) in \(P\), see [27]. In what follows by a subposet we always mean a weak subposet. The height\(h(P)\) of a poset \(P\) is the length of the longest chain in \(P\). We consider a family \(\fancyscript{F}\) of subsets of \([n]\) a poset for the subset relation. If \(P\) is not a subposet of \( \fancyscript{F}\), we say \(\fancyscript{F}\) is \(P\)-free. We are interested in determining the largest size of a \(P\)-free family of subsets of \([n]\), denoted Open image in new window. Let \(P_k\) denote the total order of \(k\) elements that we term as a \(k\)-chain. The archetypal result is Sperner’s theorem [10, 26]: Open image in new window. Let \(\fancyscript{B}(n, k)\) denote the middle \(k\) levels in the subset lattice of \([n]\) and let \(\sum (n, k) = |\fancyscript{B}(n, k)|\). Erdős [10, 11] proved that Open image in new window. \(A\nsubseteq B\)

For any family \(\fancyscript{F}\) of subsets of \([n]\), define its Lubell function\(h_n(\fancyscript{F})=\sum _{F\in \fancyscript{F}} 1/\left( {\begin{array}{c}n\\ |F|\end{array}}\right) \). The celebrated Bollobás–Lubell–Meshalkin–Yamamoto (BLYM) inequality asserts that for a \(P_k\)-free family \(\fancyscript{F}\), we have \(h_n(\fancyscript{F}) \le k-1\), which was originally shown for \(k=2\) [4, 22, 24, 29] and extended for all \(k\) by Erdős, Füredi, and Katona [12]. (For a generalization of the BLYM inequality, where cases of equality characterize mixed orthogonal arrays, see Aydinian, Czabarka, and Székely [2].) The BLYM inequality gives the book proof to Open image in new window. In view of this, it makes sense to study \(\lambda _n(P) = \max {h_n(\fancyscript{F})}\), where the maximization is over \(P\)-free families \(\fancyscript{F}\) in \([n]\).

Katona had a key role starting the investigation of extremal problems with excluded posets [6, 7, 8, 13, 18]. Katona and Tarján [18] obtained bounds on Open image in new window and later De Bonis and Katona [7] extended it to Open image in new window, where the \(r\)-fork \(V_r\) is the poset \(A < B_1, \ldots , B_r\). The answers are asymptotic to \(\left( {\begin{array}{c}n\\ \lfloor n/2\rfloor \end{array}}\right) \), and most of the work is devoted to finding second and third terms in the asymptotic expansion.

For excluded posets \(P\) whose Hasse diagram is a tree, surpassing earlier results of Thanh [28] and Griggs and Lu [16], finally Bukh [5] solved the asymptotic problem for the main term. For any poset \(P\), define \(e(P)\) to be the maximum \(m\) such that for all \(n\), the union of the \(m\) middle levels \(\fancyscript{B}(n,m)\) does not contain \(P\) as a subposet. The relevance of \(e(P)\) was suggested by Saks and Winkler. Bukh [5] showedis \(e(P)\).

Katona [17] attributes the now famous diamond problem to a question of an unidentified member of the audience at his talk. After all, if trees are solved, the next open problems must allow some cycles in the Hasse diagram of \(P\). The diamond\(D_k\) is defined as \(A < B_1\), \(\ldots \), \(B_k < C\), and with the term diamond we normally refer to \(D_2\).

Griggs and Li [14] introduced a relevant class of posets. They termed a poset \(P\)uniform-L-bounded if \(\lambda _n(P) \le e(P)\) for all \(n\). For any uniform-L-bounded posets \(P\), Griggs and Li [14] proved Open image in new window. Griggs, Li, and Lu [15] showed that the chain \(P_k\), the diamond \(D_k\) if \(2^{m-1}-1\le k \le 2^m- \left( {\begin{array}{c}m\\ \lfloor m/2\rfloor \end{array}}\right) -1\) (including the numbers \(k = 3, 4, 7, 8, 9, 15, 16, \ldots \)) are uniform-L-bounded posets, and so are the harps \(H(l_1, l_2, \ldots , l_k)\) (consisting of chains \(P_1,\ldots ,P_k\) with their top elements identified and their bottom elements identified) for \(l_1 > l_2 > \cdots > l_k\).

Griggs and Lu [16] conjecture that for any poset\(P\), the limit in (1) exists and is an integer. All the results above are compatible with this conjecture.

The crown\({\fancyscript{O}}_{2t}\) is defined as \(A_1<B_1>A_2<B_2> \cdots >A_t<B_t>A_1\). Crowns are neither trees nor uniform-L-bounded. The crown \({\fancyscript{O}}_{4}\) is known as the butterfly. De Bonis et al.  [8] proved Open image in new window. Griggs and Lu [16] proved \(\pi ({\fancyscript{O}}_{2t})=e({\fancyscript{O}}_{2t})\) for \(t\ge 4 \) even, and recently Lu [20] proved \(\pi ({\fancyscript{O}}_{2t})=e({\fancyscript{O}}_{2t})\) for \(t\ge 7 \) odd, leaving open only \({\fancyscript{O}}_{6}\) and \({\fancyscript{O}}_{10}\).

The most famous open problem about excluded posets is that of the diamond \(D_2\). Griggs and Lu showed [16]The cited conjecture of Griggs and Lu would imply that in (2) the limit exists and is equal to \(2\). This is what we refer to as the diamond conjecture. Axenovich et al. [1] reduced the upper bound in (2) to \(2.283\); Griggs et al. [15] further reduced it to \(25/11\). The current best upper bound is \(2.25\), achieved by Kramer et al. [19]. They also pointed out that this is the best possible bound that can be derived from a Lubell function argument. Manske and Shen [23] have a better upper bound for \(3\)-layered families of sets, \(2.1547\), improving on an earlier bound of Axenovich et al. [1], \(2.207\). A similar improvement for \(3\)-layered families of sets was also made by Balogh et al. [3].

For a long time no construction was known for the diamond problem with more sets than those in the two largest levels, though some alternative constructions existed, e.g. taking on \(12\) points all \(5\)-subsets, all \(7\)-subsets and a Steiner system \(S(5,6,12)\). In 2013, Dove [9] made an improvement on this, for every even \(n\ge 6\). For \(n=6\), he provided \(36\) sets, while the two largest levels contain only \(35\) sets. As \(n\) goes to infinity, the gain in his construction is diminishing in percentage. Through computer search, Lu independently found a construction for \(n=6\) with \(36\) sets [21].

The goal of this paper is to provide infinitely many exotic examples that show the asymptotical tightness of the diamond conjecture. These constructions are based on Abelian groups and are very different from the usual extremal set systems. We show that Dove’s example fits into this description although his formulation was different. The proofs use the theory of Markov chains on groups, allowing citations of theorems instead of making analytic proofs from scratch to show a limiting uniform distribution. Dove’s example, however, uses a non-uniform distribution.

2 Strongly diamond-free Cayley posets

Let us be given a finite group \({\Gamma }\) with identity \(e\) and a set of generators \(H\subseteq {\Gamma }\). Recall that the Cayley graph\(\mathbf {G}({\Gamma }, H)\) has vertex set \({\Gamma }\) and edge set \(\{g \rightarrow gh: h\in H,g\in {\Gamma } \}\). We do not assume \(H=H^{-1}\), an assumption often made for Cayley graphs. We define the infinite Cayley poset\(P({\Gamma }, H)\) as follows: the vertices of the poset are ordered pairs \((\gamma , i)\), for \(\gamma \in {\Gamma }\) and \(i\in \mathbb Z\), and \((\gamma , i)\preceq _P (\delta ,j)\) if \(j\ge i\) and \(\gamma =\delta \eta _1\eta _2\cdots \eta _{j-i}\) for some \(\eta _1,\eta _2,\ldots ,\eta _{j-i}\in H\). It is easy to see that \(P({\Gamma }, H)\) is a partial order indeed. Furthermore, mapping the vertices of the infinite Cayley poset \(P({\Gamma }, H)\) to the vertices of the Cayley graph \(\mathbf {G}({\Gamma }, H)\) by projection to the first coordinate, upward oriented edges of the Hasse diagram map to the edges of the Cayley graph. We term finite subposets of the infinite Cayley poset as Cayley posets.

We say that \(H\) is aperiodic, if forthe greatest common divisor of elements of \(L\) is 1. We say that a (finite) Cayley poset is aperiodic if the generating set is aperiodic.
Assume now that \({\Gamma }\) is Abelian of order \(m\) and \(|H|=h\) with \(H=\{\eta _1,\eta _2,\ldots , \eta _h\}\). For convenience, for Abelian groups we use the additive notation. Let us be given an \(n\)-element set \(N\) partitioned into classes \(N_1,N_2,\ldots ,N_h\) such that \(|N_i|=n_i\). Assign for \(x\in N\) a weight\(w(x)\in H\) such that for all \(x\in N_i\), \(w(x)= \eta _i\). We will refer to \(N\) as a weighted set. For \(A\subseteq N\), define \(w(A)=\sum _{x\in A} w(x)\). For every \(i\ge 0\) and \(\gamma \in {\Gamma }\), defineLet \((\gamma _1, i_1)\prec (\gamma _2, i_2)\prec (\gamma _3, i_3)\) be three distinct elements of a Cayley poset \({\Pi }\). If some \(\eta \in H\) can be used in both an Open image in new window-term sum of elements of \(H\) representing \(\gamma _2-\gamma _1\) and an Open image in new window-term sum of elements of \(H\) representing \(\gamma _3-\gamma _2\), we say that the three elements form a strong chain in \({\Pi }\). We call a Cayley poset \({\Pi }\)strongly diamond-free if (a) \({\Pi }\) is diamond-free, and (b) it has no strong chains. We need the following easy lemma.

Lemma 2.1

If a Cayley poset \({\Pi }\) with elements Open image in new window is strongly diamond-free, then for a weighted \(n\)-element set \(N\), the family of setsis diamond-free.

Proof

Referring to a diamond in this proof, we assume that \(a_1\) is its lowest element, \(a_4\) is its largest element, and \(a_2,a_3\) are the middle (incomparable) elements. We will show that if \(\fancyscript{F}(N,w,{\Pi }) \) is not diamond-free, then \({\Pi }\) is not strongly diamond-free.

If there are four different sets \(A_1,A_2,A_3,A_4\) in \( \fancyscript{F}(N,w,{\Pi }) \) that correspond to a diamond \(a_1,a_2,a_3,a_4\), respectively, then, \(j_1< j_2,j_3< j_4\) and \(j_i=|A_i|- {\lfloor n/2\rfloor }\), and also \(\gamma _i=w(A_i)\). Now we have that either (\(j_2\ne j_3\)) or (\(j_2=j_3\) and \(\gamma _2\ne \gamma _3\)) or \((j_2=j_3\) and \(\gamma _2=\gamma _3)\).

If (\(j_2\ne j_3\)) or (\(j_2=j_3\) and \(\gamma _2\ne \gamma _3\)), then the four elements \((\gamma _i,j_i)\), \(i\in [4]\), form a diamond in \({\Pi }\) so \({\Pi }\) is not strongly diamond-free.

If \(j=j_2=j_3\) and \(\gamma =\gamma _2=\gamma _3\), then we have that \(w(A_1)=\gamma _1\), \(w(A_4)=\gamma _4\), \(|A_1|=\lfloor n/2 \rfloor +j_1\), \(|A_4|=\lfloor n/2 \rfloor +j_4\), \(w(A_2)=w(A_3)=\gamma \) and \(|A_2|=|A_3|=\lfloor n/2 \rfloor +j\).

Now clearly for \(i\in \{2,3\}\) we have that \(A_1\subseteq A_2\cap A_3\subsetneq A_i\subsetneq A_2\cup A_3\subseteq A_4\). Using Open image in new window (possibly empty), Open image in new window (possibly empty) and Open image in new window (nonempty!) for \(i\in \{2,3\}\) it follows that for Open image in new window we have that Open image in new window.

It follows that \(j=j_1+s+r\) and \(j_4=j+r+k=j_1+2r+s+k\) and that
$$\begin{aligned} \gamma =\gamma _1+\sum _{\ell =1}^s w(x_{\ell }) +\sum _{\ell =1}^r w\bigl (z^2_{\ell }\bigr ),\qquad \gamma _4=\gamma +\sum _{\ell =1}^k w(y_{\ell }) +\sum _{\ell =1}^r w\bigl (z^2_{\ell }\bigr ), \end{aligned}$$
where \(j-j_1=s+r\) and \(j_4-j=k+r\). Now since \(r\ge 1\), we can choose \(\eta =w\bigl (z_1^2\bigr )\).

In particular, in this case we have found \((\gamma _1,j_1)<(\gamma ,j)<(\gamma _4,j_4)\) in \({\Pi }\) such that \(\gamma -\gamma _1\) can be written as a Open image in new window-term sum of elements of \(H\) containing the term \(\eta \) and and \(\gamma _4-\gamma \) is a Open image in new window-term sum of elements of \(H\) containing the term \(\eta \). Thus, in this case \({\Pi }\) is not strongly diamond-free either.\(\square \)

3 Markov chains on \({\Gamma }\)

Let us be given the set \(N=[1,n]\). Assign i.i.d. \(H\)-valued random variables \(\omega (x)\) for every \(x\in N\). Assume Open image in new window for all \(\eta \in H\) and extend the probability distribution to Open image in new window by Open image in new window. For an arbitrary \(A\subseteq N\), assume Open image in new window. Now we associate a finite Markov chain \(X^A_j\) on \({\Gamma }\) for \(j=0,1,\ldots ,|A|\) with \(A\): define it with \(X_0^A=0\) for sure, and
$$\begin{aligned} X_i^A=\gamma \qquad \text {iff}\qquad \text {there exists }\delta \in {\Gamma } \text { such that } X_{i-1}^A=\delta \text { and } \omega (a_{i})=\gamma -\delta . \end{aligned}$$
More explicitly, \(X_i^A=\omega (a_1)+\omega (a_2)+\cdots + \omega (a_i)\). Consequently,If we defined analogously the infinite Markov chain \(X^A_j\) on \({\Gamma }\) for \(j=0,1,\ldots \) and an infinite \(A\subseteq \mathbb N\), the Markov chain would be irreducible if and only if \(H\) is a generating set, and in this case the Markov chain would be aperiodic if and only if \(H\) is not contained in a coset of a proper normal subgroup of \({\Gamma }\), see [25, Proposition 2.3]. If the Markov chain is irreducible and aperiodic, then it converges to the unique stationary distribution on \({\Gamma }\), which is the uniform distribution, see [25, p. 271]. Hence assuming that \(H\) is an aperiodic generating set, it gives rise to an aperiodic Markov chain, and \(X^A_j\) converges to the uniform distribution. The same results hold as well for \(X^A_j\) for a finite set \(A\), if \(|A|\) is sufficiently large for a fixed \({\Gamma }\).
The Markov chains \(X^A\) with different sets \(A\) do correlate, but we only will use the linearity of expectation. Define \(\omega (A)=\sum _{x\in A} \omega (x)\). For a fixed \(i\) and a large \(n\), seta random family of sets. Note that \(\omega (A)=X_{|A|}^A \) and \(S_\gamma (i)\) are random variables, unlike \(w(A)\) and \(s_\gamma (i)\) in the previous section. By the convergence to uniform distribution recalled above, we have that for all \(\epsilon >0\) and all sufficiently large \(n\)HenceWe reformulate (3) above as a theorem

Theorem 3.1

(equidistribution theorem) Assume that \(H\) is an aperiodic generating set of a finite Abelian group of order \(m\). Under the model above, for \(i\) fixed as \(n\rightarrow \infty \), we have
Observe that Open image in new window, when \({\Pi }\) is strongly diamond-free. Combining Lemma 2.1, Theorem 3.1, and the fact that for \(i\) fixed, the asymptotic formula
$$\begin{aligned} {\left( {\begin{array}{c}n\\ \lfloor n/2\rfloor + i\end{array}}\right) }\sim {\left( {\begin{array}{c}n\\ \lfloor n/2\rfloor \end{array}}\right) } \end{aligned}$$
holds as \(n\rightarrow \infty \), we immediately obtain the following theorem

Theorem 3.2

Assume that \(H\) is an aperiodic generating set of a finite Abelian group of order \(m\). If a fixed Cayley poset \({\Pi }\) with elements \((\gamma _i, j_i)\), \(i=1,2,\ldots ,\ell \), is strongly diamond-free, then

The conclusion of this theorem is that if one constructs an aperiodic generating set of a finite Abelian group of order \(m\) and strongly diamond-free Cayley poset of \(\ell \) elements with this generating set, then for a large \(n\), \(\fancyscript{F}(N,w,{\Pi })\) with some weighting \(w\) has size at least \(( \ell /m-\epsilon ){\left( {\begin{array}{c}n\\ \lfloor n/2\rfloor \end{array}}\right) }\). A construction with \(\ell >2m\) would even refute the diamond conjecture.

Unfortunately, the \(\ell /m\) lower bound of Theorem 3.2 never exceeds two. The proof is the following. Take any \(\eta \in H\) and partition the infinite Cayley poset \(P({\Gamma }, H)\) into \(|{\Gamma }|\) chains asIf a finite Cayley poset \({\Pi }\) is free of strong chains, which is part of the requirement to be strongly diamond-free, then \({\Pi }\) cannot have more than two elements from any of the Open image in new window, for any \(\gamma \).

Therefore in the next section we focus on constructing finite Cayley posets with \(\ell =2m\) or with just slightly fewer elements.

4 Constructions

Example 4.1

(classic example) For any \({\Gamma }\) and \(H\), take two levels from the infinite Cayley poset. This is a strongly diamond-free Cayley poset with \(2m\) vertices.

We leave the verification of the correctness of the following constructions to the readers, where \({\Gamma }\) denotes the additive group of modulo \(m\) residue classes.

Example 4.2

For an odd \(m\), take \({\Gamma }=\mathbb Z_m\) with \(H=\{a,b\}\) such that Open image in new window. The following is a strongly diamond-free Cayley poset with \(2m\) vertices:
$$\begin{aligned} (g, 3): g\not \equiv a+b \hbox { mod } m,\quad (a,2),\quad (b,2),\quad (g, 1): g\not \equiv 0 \hbox { mod } m. \end{aligned}$$

(Note that if Open image in new window, then \(H=\{a,b\}\) is a set of generators.) This poset is often aperiodic. For example, if \(m=3\) and the generators are \(a=1\), \(b=2\), we have an aperiodic poset. The exact condition for aperiodicity is that \(H\) is not contained by a coset of a proper subgroup of \({\Gamma }\). See [25, Proposition 2.3].

Example 4.3

Take \({\Gamma }=\mathbb Z_7\) with \(H=\{2,3,5\}\). The following is a strongly diamond-free, aperiodic Cayley poset with \(13\) vertices:
$$\begin{aligned} (g, 3): g\not \equiv 0,1,5 \hbox { mod } 7,\quad (2,2),\quad (3,2),\quad (5,2), \quad (g, 1): g\not \equiv 0 \hbox { mod } 7. \end{aligned}$$

Note that \(1,2\,\mathrm{mod}\,3\) and \(2,3,5\,\mathrm{mod}\, 7\) are difference sets. However, bigger difference sets do not seem to offer good constructions. On four levels, we still can construct “close” constructions.

Example 4.4

For \(m=4k-1\), take \({\Gamma }=\mathbb Z_m\) with \(H=\{2k-1,2k\}\), \(k\ge 2\). The following is a strongly diamond-free, aperiodic Cayley poset with \(2m-2\) vertices:
$$\begin{aligned} (i, 4): i=k+2,\ldots ,3k-3,\quad (i, j): i=k,k+1,\ldots ,3k-1, j=1,2,3. \end{aligned}$$

Example 4.5

For \(m=4k+1\), take \({\Gamma }=\mathbb Z_m\) with \(H=\{2k,2k+1\}\), \(k\ge 2\). The following is a strongly diamond-free, aperiodic Cayley poset with \(2m-2\) vertices:
$$\begin{aligned} (i, 4): i=k+2,\ldots ,3k-2,\quad (i, j): i=k,k+1,\ldots ,3k,j=1,2,3. \end{aligned}$$

The last two constructions still allow close approximations of the conjectured maximum \((2+o(1)){\left( {\begin{array}{c}n\\ \lfloor n/2\rfloor \end{array}}\right) }\). For any fixed \(\epsilon \), set \(k>1+1/\epsilon \) to have \((2m-2)/m>2-\epsilon /2\). Fixing this \(k\), for sufficiently large \(n\), a set system is obtained from Example 4.4 or 4.5 with at least \((2-\epsilon ){\left( {\begin{array}{c}n\\ \lfloor n/2\rfloor \end{array}}\right) }\) elements.

Dove’s construction for even \(n\ge 6\) is the following: take for the underlying set \(N=\{1,2,\ldots ,n\}\), select those Open image in new window-element sets that do not contain the set \(\{1,2\}\), select those \(\lfloor n/2 \rfloor \)-element sets that do not contain exactly one element from the set \(\{1,2\}\), and select those Open image in new window-element sets that have non-empty intersection with the set \(\{1,2\}\). It is easy to check that this family is diamond-free. Say, for \(n=6\), the family has
$$\begin{aligned} \Biggl [\left( {\begin{array}{c}6\\ 2\end{array}}\right) -1\Biggl ]+\Biggl [\left( {\begin{array}{c}4\\ 3\end{array}}\right) +\left( {\begin{array}{c}4\\ 1\end{array}}\right) \Biggl ] +\Biggl [\left( {\begin{array}{c}6\\ 4\end{array}}\right) -1\Biggl ]=14+8+14=36 \end{aligned}$$
elements, while two distinct middle levels have only \(\left( {\begin{array}{c}6\\ 2\end{array}}\right) + \left( {\begin{array}{c}6\\ 3\end{array}}\right) =15+20=35 \) elements.

Example 4.6

Take \({\Gamma }=\mathbb Z_3\) with \(H=\{0,2\}\). The following is a strongly diamond-free, aperiodic Cayley poset with six vertices:
$$\begin{aligned} (g, 1): g\not \equiv 1 \hbox { mod } 3,\quad (g, 0): g\not \equiv 0 \hbox { mod } 3,\quad (g, -1): g\not \equiv 2 \hbox { mod } 3. \end{aligned}$$

Dove’s construction can be obtained from this example using Lemma 2.1 but not Markov chains. Take for the underlying set \(N=\{1,2,\ldots ,n\}\), set \(N_1=\{1,2\} \) with weights \(w(1)=w(2)=2\in \mathbb Z_3\) and \(N_2=\{3,4,\ldots ,n\}\) with weights \(w(3)=w(4)=\ldots =w(n)=0\in \mathbb Z_3\). This construction gives the very same family as Dove’s construction.

Notes

Acknowledgments

The authors thank the referees for their comments and corrections.

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Copyright information

© Springer International Publishing AG 2015

Authors and Affiliations

  • Éva Czabarka
    • 1
  • Aaron Dutle
    • 2
  • Travis Johnston
    • 3
  • László A. Székely
    • 1
  1. 1.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  2. 2.Safety Critical Avionics Systems BranchNASA Langley Research CenterHamptonUSA
  3. 3.Department of Computer and Information SciencesUniversity of DelawareNewarkUSA

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