Structure and automorphisms of primitive coherent configurations
Coherent configurations (CCs) are highly regular colorings of the set of ordered pairs of a “vertex set”; each color represents a “constituent digraph.” CCs arise in the study of permutation groups, combinatorial structures such as partially balanced designs, and the analysis of algorithms; their history goes back to Schur in the 1930s. A CC is primitive (PCC) if all its constituent digraphs are connected.
We address the problem of classifying PCCs with large automorphism groups. This project was started in Babai’s 1981 paper in which he showed that only the trivial PCC admits more than automorphisms. (Here, is the number of vertices and the hides polylogarithmic factors.)
In the present paper we classify all PCCs with more than automorphisms, making the first progress on Babai’s conjectured classification of all PCCs with more than automorphisms.
A corollary to Babai’s 1981 result solved a then 100-year-old problem on primitive but not doubly transitive permutation groups, giving an bound on their order. In a similar vein, our result implies an upper bound on the order of such groups, with known exceptions. This improvement of Babai’s result was previously known only through the Classification of Finite Simple Groups (Cameron, 1981), while our proof, like Babai’s, is elementary and almost purely combinatorial.
Our analysis relies on a new combinatorial structure theory we develop for PCCs. In particular, we demonstrate the presence of “asymptotically uniform clique geometries” on PCCs in a certain range of the parameters.
Let be a finite set; we call the elements of “vertices.” A configuration of rank is a coloring such that (i) for any , and (ii) for all there is such that iff . The configuration is coherent (CC) if (iii) for all there is a structure constant such that if , there are exactly vertices such that and . The diagonal colors are the vertex colors, and the off-diagonal colors are the edge colors. A CC is homogeneous (HCC) if (iv) there is only one vertex color. We denote by the set of ordered pairs of color . The directed graph is the color- constituent digraph. An HCC is primitive (PCC) if each constituent digraph is strongly connected. An association scheme is an HCC for which for all colors (so the constituent graphs are viewed as undirected).
The term “coherent configuration” was coined by Donald Higman in 1969 , but the essential objects are older. In the case corresponding to a permutation group, CCs already effectively appeared in Schur’s 1933 paper . This group-theoretic perspective on CCs was developed further by Wielandt .
CCs appeared for the first time from a combinatorial perspective in a 1952 paper by Bose and Shimamoto . They, along with many of the subsequent authors, consider the case of an association scheme, which is essential for understanding partially balanced incomplete block designs, of interest to statisticians and to combinatorial design theorists. The generalization of an association scheme to an HCC was considered by Nair in 1964 . The algebra associated with a CC, which already appeared in Schur’s paper, was rediscovered in 1959 in the context of association schemes by Bose and Mesner .
Weisfeiler and Leman  and Higman  independently defined CCs in their full generality, including the associated algebra (called “cellular algebras” by Weisfeiler and Leman), in the late 1960s. For Higman, CCs were a generalization of permutation groups, whereas Weisfeiler and Leman were motivated by the algorithmic Graph Isomorphism problem. In the intervening years, CCs, and association schemes in particular, have become basic objects of study in algebraic combinatorics [8, 11, 7, 29]. CCs also continue to play a role in the study of permutation groups [16, 20]. Recent algorithmic applications of the CC concept include the Graph Isomorphism problem  and the complexity of matrix multiplication .
PCCs are in a sense the “indivisible objects” among CCs and are therefore of particular interest.
In this paper we classify the PCCs with the largest automorphism groups, up to the threshold stated in the following theorem. (See Defintion 1.3 and Theorem 1.4 for a more detailed statement, and see Section 1.6 for an explanation of the asymptotic notation used throughout, including , , , , , and .)
If is a PCC not belonging to any of three exceptional families, then .
Primitive permutation groups of large order were classified by Cameron . We refer to the orbital, or Schurian configurations of these groups as “Cameron schemes” (see Sections 1.1 and 1.2). For every , for every there is only a bounded number of primitive groups of order greater than (the bound depends on but not on ); we refer to this stratification as the “Cameron hierarchy.”
Theorem 1.1 represents progress on the following conjectured classification of PCCs with large automorphism groups.
Conjecture 1.2 (Babai).
For every , there is some such that if is a PCC on vertices and , then is a Cameron scheme. In particular, is a known primitive group.
This conjecture would be a far-reaching combinatorial generalization of Cameron’s classification of large primitive permutation groups. In particular, while Cameron’s result is only known through the Classification of Finite Simple Groups (CFSG), Conjecture 1.2 would imply (at least for orders greater than ) a CFSG-free proof of Cameron’s result, giving a different kind of insight into the structure of large primitive permutation groups.
Babai  established Conjecture 1.2 for all (the “first level of the Cameron hierarchy”). As a corollary, he solved a then 100-year-old problem on primitive but not doubly transitive permutation groups, giving an nearly tight, bound on their order. The tight bound was subsequently found by Cameron, using CFSG; our result implies a CFSG-free proof of the same tight bound. Moreover, our Theorem 1.1 confirms the conjecture to all , the first improvement since Babai’s paper. An elementary proof of an upper bound on the order of primitive permutation groups, with known exceptions (the “second level of the Cameron hierarchy”) follows.
For the proof of Theorem 1.1, we find new combinatorial structure in PCCs, including “clique geometries” in certain parameter ranges (Theorem 2.4). An overview of our structural results for PCCs is given in Section 2.
Our motivation is thus twofold. First, we develop a structure theory for PCCs, the most general objects in a hierarchy of much-studied highly regular combinatorial structures. Second, as a corollary to our main result, we obtain a CFSG-free proof for the second level of the Cameron hierarchy of large primitive permutation groups.
Additional motivation for our work comes from the algorithmic Graph Isomorphism problem. We explain this connection in Section 1.5.
1.1 Exceptional coherent configurations
We now give a precise statement of our main combinatorial results.
Given a graph , we associate with the configuration where denotes the set of edges of the complement of . (We omit if and omit if .) So graphs can be viewed as configurations of rank .
Given an (undirected) graph , the line-graph has as vertices the edges of , with two vertices adjacent in if the corresponding edges are incident in . The triangular graph is the line-graph of the complete graph (so ). The lattice graph is the line-graph of the complete bipartite graph (on equal parts) (so ). The configurations and are coherent, and in fact primitive for .
A PCC is exceptional if it is of the form , where is isomorphic to the complete graph , the triangular graph , or the lattice graph , or the complement of such a graph.
We note that the exceptional PCCs have automorphisms. Indeed, the exceptional PCCs are exactly the “orbital configurations” of large primitive permutation groups, as explained below.
Our main result is that all the non-exceptional PCCs have far fewer automorphisms.
If is a non-exceptional PCC, then .
We remark that the bound of Theorem 1.4 is tight, up to polylogarithmic factors in the exponent. Indeed, the Johnson scheme and the Hamming scheme both have automorphisms. (The Johnson scheme has vertices the -subsets of a domain of size and , and the Hamming scheme has vertices the words of length from an alphabet of size with color given by the Hamming distance.)
The exceptional PCCs correspond naturally to the largest primitive permutation groups. Given a permutation group , we define the orbital configuration on vertex set with the given by the orbitals of , i.e., the orbits of the induced action on . CCs of this form were first considered by Schur , and are commonly called Schurian. Note that .
The Schurian CC is homogeneous if and only if is transitive, and primitive if and only if is a primitive permutation group. If is doubly transitive, then . We also have and .
1.2 Primitive permutation groups
Following the completion of the Classification of Finite Simple Groups (CFSG), one of the tasks has been to obtain elementary proofs of results currently known only through CFSG. One such result is Cameron’s classification of all primitive permutation groups of large order, obtained by combining CFSG with the O’Nan–Scott theorem . Cameron’s threshold for the order is , but we state Maróti’s refinement of his classification of permutation groups of order greater than .
Theorem 1.5 (Cameron, Maróti).
If is a primitive permutation group of degree , then one of the following holds:
there are positive integers , , and such that ;
We call the primitive groups of Theorem 1.5 (a) Cameron groups. Given a Cameron group with parameters and bounded, we obtain a PCC with exponentially large automorphism group , in particular, of order . We call the PCCs Cameron schemes when is a Cameron group.
Hence, Conjecture 1.2 states that Cameron’s classification of primitive permutation groups transfers to the combinatorial setting of PCCs. Furthermore, the conjecture entails Cameron’s theorem, above the threshold (see ). Hence, confirmation of Conjecture 1.2 would yield a CFSG-free proof of Cameron’s classification (above this threshold).
It seems unlikely that combinatorial methods will match Cameron’s threshold for classification of primitive permutation groups. An threshold (as in stated Theorem 1.5) via elementary techniques might be possible, since above this threshold the socle of a primitive permutation group is a direct product of alternating groups, whereas below this threshold, simple groups of Lie type may appear in the socle.
However, until the present paper, the only CFSG-free classification of the large primitive permutation groups was given by Babai in a pair of papers in 1981 and 1982 [1, 2]. Babai proved that for primitive groups other than and . A corollary of our work gives the first CFSG-free improvement to Babai’s bound, by proving that for primitive permutation groups , other than groups belonging to three exceptional families.
In the following corollary to Theorem 1.1, and denote the actions of and , respectively, on the pairs, and denotes the wreath product of the permutation groups by in the product action on a domain of size .
Let be a primitive permutation group of degree . Then either , or is one of the following groups:
or , where ;
a subgroup of containing , where .
The slightly stronger bound follows from CFSG . By contrast, our proof is elementary.
For given , there are exactly three primitive groups in the third category of Corollary 1.6. We note that the groups of categories 1–3 of the corollary have order .
1.3 Individualization and refinement
We now introduce the individualization/refinement heuristic. We shall use individualization/refinement to find bases of automorphism groups of configurations.
A base for a group acting on a set is a subset such that the pointwise stablizer of in is trivial. If is a base, then .
Let denote the set of isomorphisms from to , and .
Individualization means the assignment of individual colors to some vertices; then the irregularity so created propagates via some canonical color refinement process. For a class of configurations (not necessarily coherent), an assignment is a color refinement if have the same set of vertices and the coloring of is a refinement of the coloring of . Such an assignment is canonical if for all , we have . In particular, .
Repeated application of the refinement process leads to the stable refinement after at most rounds.
If after individualizing the elements of a set , all vertices get different colors in the resulting stable refinement, we say that completely splits (with respect to the given canonical refinement process). If completely splits , then is a base for . Hence, to prove Theorem 1.4, it suffices to show that some set of vertices completely splits after canonical color refinement.
For our purposes, the simple “naive vertex refinement” will suffice as our color refinement procedure. Under naive vertex refinement, the edge-colors do not change, only the vertex-colors are refined. The refined color of vertex of the configuration encodes the following information: the current color of and the number of vertices of color such that , for every pair , where is a vertex-color and is an edge-color.
We now state our main technical result, from which Theorem 1.4 immediately follows.
Theorem 1.8 (Main).
Let be a non-exceptional PCC. Then there exists a set of vertices that completely splits under naive refinement.
This improves the main result of , which stated that if is a PCC other than , then there is a set of vertices which completely splits under naive refinement.
Naive vertex refinement is the only color refinement used in the present paper. However, we remark that coherent configurations were first studied by Weisfeiler and Leman in the context of their stronger canonical color refinement [27, 26].
Given a configuration , the Weisfeiler–Leman (WL) canonical refinement process [27, 26] produces a CC on the same vertex set with , by refining the coloring until it is coherent. More precisely, in every round of the refinement process, the color of the pair is replaced with a color which encodes along with, for every pair of original colors, the number of vertices such that and . This refinement is iterated until the coloring stabilizes, i.e., the rank no longer increases in subsequent rounds of refinement. The stable configurations under WL refinement are exactly the coherent configurations.
1.4 Relation to strongly regular graphs
An undirected graph is called strongly regular (SRG) with parameters if has vertices, every vertex has degree , each pair of adjacent vertices has common neighbors, and each pair of non-adjacent vertices has common neighbors.
We note that a graph is a SRG if and only if the configuration is coherent. If a SRG is nontrivial, i.e., it is connected and coconnected, then is a PCC.
All of our exceptional PCCs are in fact SRGs. Our classification of PCCs, Theorem 1.4, was established in the special case of SRGs by Spielman in 1996 , on whose results we build. In fact, Chen, Sun, and Teng have now established a stronger bound for SRGs: a non-exceptional SRG has at most automorphisms .
The results of Spielman and Chen, Sun, and Teng both rely on Neumaier’s structure theory  of SRGs to separate the exceptional SRGs with many automorphisms from those to which I/R can be effectively applied. However, no generalization of Neumaier’s results to PCCs has been known. We provide a weak generalization, sufficient for our purposes, in Section 2.
1.5 Graph Isomorphism
The “Graph Isomorphism (GI) problem” is the computational problem to decide whether or not a pair of given graphs are isomorphic. This problem is of great interest to complexity theory since it is one of a very small number of natural problems in NP of intermediate complexity status (unlikely to be NP-complete but not known to be solvable in polynomial time).
In recent major development, Babai  announced a quasipolynomial-time () algorithm.
Babai’s algorithm reduces the problem to the isomorphism problem of PCC’s and then uses his (rather involved) “split-or-Johnson” procedure for further reduction.
Babai conjectures that a considerably simpler algorithm might succeed; unless the PCC is a Cameron scheme, individualization of a small number of vertices may completely split the vertex set. This is a more explicit version of Conjecture 1.2.
Our result proves that this is indeed the case if “small number” means , improving Babai’s . We hope that further refinement of our structure theory will yield further progress in this direction.
1.6 Asymptotic notation
To interpret asymptotic inequalities involving the parameters of a PCC, we think of the PCC as belonging to an infinite family in which the asymptotic inequalities hold.
For functions , we write if there is some constant such that , and we write if . We write if and . We use the notation when there is some constant such that . We write if for every , there is some such that for , we have . We write if . We use the notation for asymptotic equality, i.e., . The asymptotic inequality means .
The authors are grateful to László Babai for sparking our interest in the problem addressed in this paper, providing insight into primitive coherent configurations and primitive groups, uncovering a faulty application of previous results in an early version of the paper, and giving invaluable assistance in framing the results.
2 Structure theory of primitive coherent configurations
To prove Theorem 1.8, we need to develop a structure theory of PCCs. The overview in this section highlights the main components of that theory.
Throughout the paper, will denote a PCC of rank on vertex set with structure constants for . We assume throughout that , since the case is the trivial case of , listed as one of our exceptional PCCs. We also assume without loss of generality that color corresponds to the diagonal, i.e., .
For any color in a PCC, we write , the out-degree of each vertex in .
We say that color is dominant if . Colors with are nondominant. We call a pair of distinct vertices dominant (nondominant) when its color is dominant (nondominant, resp.). We say color is symmetric if . Note that when color is dominant, it is symmetric, since .
Our analysis will divide into two cases, depending on whether or not there is a dominant color. In fact, many of the results of this section will assume that there is an overwhelmingly dominant color satisfying . The reduction to this case is accomplished via Lemma 3.1 of the next section. The main structural result used in its proof is Lemma 2.1 below, which gives a lower bound on the growth of “spheres” in a PCC.
For a color and vertex , we denote by the set of vertices such that . We denote by the directed distance from to in the color- constituent digraph , and we write for any vertices with . (This latter quantity is well-defined by the coherence of .) The -sphere in centered at is the set of vertices with .
Lemma 2.1 (Growth of spheres).
Let be a PCC, let be nondiagonal colors, let , and . Then for any integer , we have
We note that Lemma 2.1 is straightforward when is distance-regular. Indeed, a significant portion of the difficulty of the lemma was in finding the correct generalization.
Overview of proof of Lemma 2.1.
The bipartite subgraphs of induced on pairs of the form , where are colors and is a vertex, are biregular by the coherence of . We exploit this biregularity to count shortest paths in between a carefully chosen subset of and , for an arbitrary vertex .
The details of the proof are given in Section 4. ∎
In the rest of the paper, we assume without loss of generality that . We write . For the rest of the section, color will in fact be dominant. In fact, every theorem in the rest of this section will state the assumption that . Lemma 2.2 below demonstrates some of the power of this supposition.
Let and let be a PCC with . Then, for sufficiently large, for every nondominant color . Consequently, for .
Overview of proof of Lemma 2.2.
We will prove that if for some color , then . Without loss of generality, we assume , since otherwise we are already done.
Fix an arbitrary vertex and consider the bipartite graph between and , with an edge from to when . By the coherence of , the bipartite graph is regular on ; call its degree . An obstacle to our analysis is that the graph need not be biregular. Nevertheless, we estimate the maximum degree of a vertex in in . We first note that .
Let be a vertex satisfying . We pass to the subgraph induced on , and observe that the degree of vertices in is preserved, while the degree of vertices in does not increase. Let be a vertex of degree in , and let . We finally consider the bipartite graph on , where is the set of vertices with at most in-neighbors in lying in the set . In particular, . This graph is now regular (of degree ) on . Since , we have , which eventually gives the bound . Combining this with our earlier estimate proves the lemma.
The details of the proof are given in Section 6. ∎
Notation. Let be the graph on formed by the nondominant pairs. So is regular of valency , and every pair of distinct nonadjacent vertices in has exactly common neighbors, where . The graph is not generally SR, since pairs of adjacent vertices in of different colors in will in general have different numbers of common neighbors. However, intuition from SRGs will prove valuable in understanding .
We write for the set of neighbors of in the graph . For nondominant, we define , where . So, the parameters are loosely analogous to the parameter of a SRG.
A clique in an undirected graph is a set of pairwise adjacent vertices; its order is the number of vertices in the set.
A clique geometry on a graph is a collection of maximal cliques such that every pair of adjacent vertices in belongs to a unique clique in . A clique geometry of a PCC is a clique geometry on . The clique geometry is asymptotically uniform (for an infinite family of PCCs) if for every , , and nondominant color , we have either or (as ).
We have the following sufficient condition for the existence of clique geometries in PCCs.
Let be a PCC satisfying , and fix a constant . If for every nondominant color , then for sufficiently large, there is a clique geometry on . Moreover, is asymptotically uniform.
Theorem 2.4 provides a powerful dichotomy for PCCs: either there is an upper bound on some parameter , or there is a clique geometry. Adapting a philosophy expressed in , we note that bounds on are useful because they limit the correlation between the -neighborhoods of two random vertices. Similar bounds on the parameter of a SRG were used in .
On the other hand, Theorem 2.4 guarantees that if all parameters are sufficiently large, the PCC has an asymptotically uniform clique geometry. This is our weak analogue of Neumaier’s geometric structure. Clique geometries offer their own dichotomy. Geometries with at most two cliques at a vertex can classified; this includes the exceptional PCCs (Theorem 2.5 below). A far more rigid structure emerges when there are at least three cliques at every vertex. In this case, we exploit the ubiquitous 3-claws (induced subgraphs) in in order to construct a set which completely splits (Lemma 3.4 (b)).
Overview of proof of Theorem 2.4.
The existence of a weaker clique structure follows from a result of Metsch . (See Lemma 7.1 below and the comments in the paragraph preceding it.) Specifically, under the hypotheses of Theorem 2.4, for every nondominant color and vertex , there is a partition of into cliques of order in . We call such a collection of cliques a local clique partition (referring to the color- neighborhood of any fixed vertex).
The challenge is to piece together these local clique partitions into a clique geometry. An obstacle is that Metsch’s cliques are cliques of , not ; that is, the edges of the cliques partitioning have nondominant colors but not in general color . In particular, for two vertices with , the clique containing in the partition of may not correspond to any of the cliques in the partition of .
We first generalize these local structures. An -local clique partition is a partition of into cliques of order . We study the maximal sets for which such -local clique partitions exist, and eventually prove that these maximal sets partition the set of nondominant colors, and the corresponding cliques are maximal in .
Finally, we prove a symmetry condition: given a nondominant pair of vertices , the maximal local clique at containing is equal to the maximal local clique at containing . This symmetry ensures the cliques form a clique geometry, and this clique geometry is asymptotically uniform by construction.
The details of the proof are given in Section 7. ∎
The case that has a clique geometry with some vertex belonging to at most two cliques includes the exceptional CCs corresponding to and . We give the following classification.
Let be a PCC such that . Suppose that has an asymptotically uniform clique geometry and a vertex belonging to at most two cliques of . Then for sufficiently large, one of the following is true:
has rank three and is isomorphic to or ;
has rank four, has a non-symmetric non-dominant color , and is isomorphic to for .
Overview of proof of Theorem 2.5.
We first use the coherence of to show that every vertex belongs to exactly two cliques of , and these cliques have order . By counting vertex-clique incidences, we then obtain the estimate . On the other hand, by Lemma 2.2, every nondominant color satisfies . Hence, there are at most nondominant colors.
Since every vertex belongs to exactly two cliques, the graph is the line-graph of a graph. If there is only one nondominant color, then is strongly regular, and therefore, for sufficiently large, is isomorphic to or . On the other hand, if there are two nondominant colors, by counting paths of length we show that must again be isomorphic to . By studying the edge-colors at the intersection of the cliques containing two distinct vertices and exploiting the coherence of , we finally eliminate the case that the two nondominant colors are symmetric.
The details of the proof are given in Section 8. ∎
3 Overview of analysis of I/R
We now give a high-level overview of how we apply our structure theory of PCCs to prove Theorem 1.8.
Most of the results highlighted in Section 2 assumed that . Hence, the first step is to reduce to this case, which we accomplish via the following lemma.
Let be a PCC. If , then there is a set of size which completely splits .
Let . We say distinguishes and if . We write for the set of vertices distinguishing and , and where . We call the distinguishing number of .
Hence, . If , then after individualizing and refining, and get different colors.
Babai observed that in order to completely split a PCC , it suffices to individualize some set of vertices, where [1, Lemma 5.4]. Thus, to prove Lemma 3.1, we show that if then for every color , we have .
The following bound on the number of large colors in a PCC becomes powerful when is small.
Let be a PCC. For any nondiagonal color , the number of colors such that is at most .
Overview of proof of Lemma 3.3.
Let be the set of colors such that , and the set of colors such that . For a set of colors, let be the total degree of the colors in .
First, we prove that , a lower bound on the total degree of colors with distinguishing number . Next, we prove a lemma that allows us to transfer estimates for the total degree of colors with small distinguishing number into estimates for the total degree of colors with low degree. Specifically, we prove that , where . Together, these two results allow us to transfer estimates on total degree between the sets and , as and increase.
The details of the proof are given in Section 5. ∎
Overview of proof of Lemma 3.1.
Fix a color . We wish to give a lower bound on . Babai observed that for any color , we have [1, Proposition 6.4 and Theorem 6.1]). Hence, we wish to give an upper bound on for some color with large.
We analyze two cases: and .
In the former case, when , we first observe that . Hence, the problem is reduced to bounding the quantity for every color . Our bound in Lemma 2.1 on the size of spheres suffices for this task since is large.
In the case that , Babai observed that the color maximizing satisfies . We partition the colors of according to their distinguishing number, by first partitioning the positive integers less than into cells of length . (Specifically, we partition the colors of so that the -th cell contains the colors satisfying , and there are cells.) Each cell of this partition is nonempty. In fact, we show that the sum of the degrees of the colors in each cell is at least .
On the other hand, Lemma 3.3 says that there are few colors satisfying , and we show that the total degree of the colors with is also small. Since each cell of the partition has degrees summing to at least , these together give an upper bound on the number of cells, and hence a lower bound on .
The details of the proof are given in Section 5. ∎
We have now reduced to the case that . Our analysis of this case is inspired by Spielman’s analysis of SRGs .
There exists a constant such that the following holds. Let be a PCC with . If satisfies either of the following conditions, then there is a set of vertices which completely splits .
There is a nondominant color such that .
For every nondominant color , we have . Furthermore, has an asymptotically uniform clique geometry such that every vertex belongs to at least three cliques of .
Overview of proof of Lemma 3.4.
We will show that if we individualize a random set of vertices, then with positive probability, every pair of distinct vertices gets different colors in the stable refinement.
Let , and fix two colors and . Generalizing a pattern studied by Spielman, we say a triple is good for and if , , and , but there exists no vertex such that and . (See Figure 3). To ensure that and get different colors in the stable refinement, it suffices to individualize two vertices for which there exists a vertex such that is good for and . We show that if there are many good triples for and , then individualizing a random set of vertices is overwhelming likely to result in the individualization of such a pair .
Condition (a) of the lemma is analogous to the asymptotic consequences of Neumaier’s claw bound used by Spielman  (cf. [6, Section 2.2]), except that the bound on does not imply a similar bound on . We show that a relatively weak bound on already suffices for Spielman’s argument to essentially go through. However, if even this weaker assumption fails, then we turn to our local clique structure for the analysis (as described in the overview of Theorem 2.4).
When condition (b) holds, we cannot argue along Spielman’s lines, and instead analyze the structural properties of our clique geometries to estimate the number of good triples.
The details of the proof are given in Section 9. ∎
By Theorem 2.4, either the hypotheses of of Lemma 3.4 are satisfied, or has an asymptotically uniform clique geometry , and some vertex belongs to at most two cliques of . Theorem 2.5 gives a characterization PCCs with the latter property: is one of the exceptional PCCs, or has rank four with a non-symmetric non-dominant color and is isomorphic to for . We handle this final case via the following lemma, proved in Section 8.
Let be a PCC satisfying Theorem 2.5 (b). Then some set of size completely splits .
We conclude this overview by observing that Theorem 1.8 follows from the above results.
Proof of Theorem 1.8.
Let be a PCC. Suppose first that . Then by Lemma 3.1, there is a set of size which completely splits .
4 Growth of spheres
In this section, we will prove Lemma 2.1, our estimate of the size of spheres in constituent digraphs.
We start from a few basic observations.
Let be a bipartite graph, and let be a partition of such that the subgraph induced on is biregular of positive valency for each . Then for any , we have
where is the set of neighbors of vertices in , i.e., .
Let . By the pigeonhole principle, there is some such that . Let be the degree of a vertex in and let be the number of neighbors in of a vertex in . We have , and . Hence,
Suppose are disjoint set of vertices. We denote by the bipartite graph between and such that there is an edge from to if . For a set of nondiagonal colors, we denote by the bipartite graph between and such that there is an edge from to if .
For any vertex , colors with , and set of nondiagonal colors, the bipartite graph is biregular.
The degree of every vertex in is . And the degree of every vertex in is . ∎
Recall our notation for the -sphere centered at in the color- constituent digraph, i.e., the set of vertices such that .
For the remainder of Section 4, we fix a PCC , a color , and a vertex . For a color and an integer , we denote by the set of vertices such that there is a vertex and a shortest path in from to passing through , i.e.,
Note that these sets are nonempty by the primitivity of , and in particular, if , then . For and an integer , we denote by the set of vertices such that there is a shortest path in from to passing through , i.e.
See Figure 1 for a graphical explanation of the notation.
Let be a color such that . Let be an integer, and let . Then
Consider the bipartite graph with
There is an edge from to if there is a shortest path from to passing through .
By the coherence of , if is nonempty for some color , then . Hence, is partitioned into sets of the form with . For such colors , by Fact 4.2, is biregular, and by the definition of , then is not an empty graph.
Therefore, the result follows by applying Proposition 4.1 with , ,