Generalized Ramsey numbers: forbidding paths with few colors
Robert A. Krueger

TL;DR
This paper studies a generalized Ramsey number function that measures the minimum colors needed in edge-colorings of complete graphs to ensure paths have a minimum number of colors, extending previous work to new graph types.
Contribution
It determines the asymptotic behavior of the function for paths on vertices for most values of path length and color count, broadening understanding beyond complete graphs.
Findings
Established asymptotics for f(K_n, P_v, q) for most v and q
Extended extremal function analysis to paths beyond complete graphs
Provided new bounds and exact values for specific cases
Abstract
Let be the minimum number of colors needed to edge-color so that every copy of is colored with at least colors. Originally posed by Erd\H{o}s and Shelah when is complete, the asymptotics of this extremal function have been extensively studied when is a complete graph or a complete balanced bipartite graph. Here we investigate this function for some other , and in particular we determine the asymptotic behavior of for almost all values of and , where is a path on vertices.
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Generalized Ramsey numbers: forbidding paths with few colors
Robert A. Krueger111Department of Mathematics, Miami University, [email protected]. Research was conducted at the REU at the City University of New York’s Baruch College, which was supported by NSF grant DMS-1710305.
(June 16, 2019)
Abstract
Let be the minimum number of colors needed to edge-color so that every copy of is colored with at least colors. Originally posed by Erdős and Shelah when is complete, the asymptotics of this extremal function have been extensively studied when is a complete graph or a complete balanced bipartite graph. Here we investigate this function for some other , and in particular we determine the asymptotic behavior of for almost all values of and , where is a path on vertices.
1 Introduction
The classical graph Ramsey problem asks, “for a given and , what is the smallest such that every edge-coloring of with colors contains a monochromatic ?” We may invert the roles of and in this question: for a given and , what is the largest such that every edge-coloring of with colors contains a monochromatic ? This is equivalent to determining the smallest such that there exists an edge-coloring of with colors such that every copy of contains at least colors. Let be the minimum number of colors needed to edge-color so that every copy of in contains at least colors.222In the literature these are called -colorings, and is typically taken to be or to ensure no difficulty in finding copies of ; see for example [1]. The classical Ramsey problem is then equivalent to determining ; as an example from [8], the Ramsey number is equivalent to and . Additionally, can be viewed as a variant of the rainbow Ramsey problem, where every copy of must receive all distinct colors (as defined in [2]). In this way studying for general bridges the gap between these problems.
The function was first introduced by Erdős and Shelah (see Section 5 of [5]), and while Elekes, Erdős, and Füredi had some preliminary results (described in Section 9 of [6]), the problem was first systematically studied by Erdős and Gyárfás [8]. They focused on the asymptotics of for large , with and fixed. They also determined the linear and quadratic thresholds of this function, that is, they determined the smallest such that (or , resp.). Axenovich, Füredi, and Mubayi [1] adapted these results to in addition to relating these problems to other results in extremal combinatorics. Many others (only a few of which are mentioned here) have studied and for between the linear and quadratic thresholds [14, 15], above the quadratic threshold [1, 13], and at the ‘polynomial’ threshold [3]. Besides two general results in [1], little has been said for when is not complete or complete bipartite.
The aim of this paper is to open up the study of for general . We consider and as fixed, determining the asymptotics of in terms of . In Section 2 we make some general observations for all , supplementing those in [1]. One of the first Ramsey results for non-complete graphs is due to Gerencsér and Gyárfás [10], who determined that every 2-coloring of has a monochromatic path of order . Inspired by this, in Section 3 we find the asymptotics of for most and , where denotes the path on vertices:
Theorem 1.1**.**
Let . The smallest for which is . The largest for which is , unless or , in which case and .
See Figure 1 for a complete summary of the results of Section 3. The only undetermined cases are when is odd and . Here we can show that is neither linear nor quadratic in , but somewhere in between. It is not clear what the correct asymptotics are for these cases, so we pose the following problem.
Problem 1.2**.**
Asymptotically determine for odd .
Note that is almost always either or (with the only notable exception mentioned above). This is in great contrast to the behavior of . Pohoata and Sheffer [11] showed that for ,
[TABLE]
and they noted that Theorem 2.1 (below) implies an upper bound of
[TABLE]
where goes to zero as grows. Thus, for each value of , the upper bound gets asymptotically close to the lower bound for sufficiently large . This means that , as a function of , attains arbitrarily many asymptotic values (even while ignoring subpolynomial factors), for sufficiently large . Similarly tight results can be obtained for the bipartite variant , using essentially the same method as [11].
Define , the number of ‘tiers’ for , to be the number of such that and differ by some polynomial factor. Above we concluded that is some unbounded function of , whereas Theorem 1.1 and the results of Section 2 show that are all at most for all and , where and is a star and a matching on edges, respectively. It would be interesting to know for what other classes of graphs is bounded for all . This may be the case when the graphs of are sufficiently sparse, perhaps if they were all trees.
1.1 Notation and Terminology
The number of edges of is denoted by , while the number of vertices of is denoted by . A copy of in is a subgraph of isomorphic to . All colorings are edge-colorings. For a color , the -degree of a vertex , denoted , is the number of edges of color incident to . The Turán number of , denoted is the largest number of edges an -vertex graph may have without containing a copy of . For a positive integer , is the disjoint union of copies of .
We employ the following asymptotic notation throughout: means that there exists and such that for all . Similarly, corresponds to . If and , then we say . All logarithms are base .
It will often be helpful to think of in terms of repeated colors. If color appears on exactly edges in a coloring of , then we say color is repeated times. We say has ‘repetitions’ or ‘repeats’ if , where color is repeated times, the sum being over all colors. Note that if has repetitions and contains exactly colors, then . With this in mind, let . For example, the number of colors required so that there are no ‘isosceles’ triangles is .
2 General Observations
A wide-reaching upper bound on is achieved with a simple application of the Lovász Local Lemma. This idea first appeared in [8] but was stated in full generality in [1]. The following bound is close to asymptotically tight in many cases when is large.
Theorem 2.1**.**
Let and . Then , or equivalently .
In the tradition of Erdős and Gyárfás [8], we ask what are the linear and quadratic thresholds for any graph . The question of the linear threshold was solved for connected in [1], whose result and proof we give here in slightly more generality for completeness.
Theorem 2.2**.**
Let be the number of connected components of , let , and let . Then and .
Proof.
The upper bound comes from Theorem 2.1.
For the lower bound, we show that if one color class is too large, we can find a copy of with at least repeats. Let be an edge-maximal spanning forest of (since may not be connected), and let be a tree on vertices which contains . Note that consists of along with more edges. It is well known that , that is, any graph on vertices with at least edges contains every tree on vertices.333Indeed, a graph on vertices with edges has average degree . Upon repeatedly deleting the vertices of degree less than , we have removed fewer than edges, and so we have a nonempty graph of minimum degree at least , into which we can embed . In fact, the Erdős-Sós conjecture states that , which Ajtai, Komlós, Simonovits, and Szemerédi have shown to be true for sufficiently large and (unpublished, see e.g., Section 6 of [9]). If some color in a coloring of contains more than edges, then we can find a monochromatic and thus a monochromatic , which has repeats. This in turn gives a copy of with at least repeats. If we require each copy of to have at most repeats, then we must use at least colors. More succinctly,
[TABLE]
∎
Note that when is connected, Theorem 2.2 gives the linear threshold as . What is the best one can do when is disconnected? The threshold of given by Theorem 2.2 is still correct for forests, since in that case . On the other hand, it is not immediately clear if the linear threshold for is or .
We now turn to the quadratic threshold, the smallest for which is quadratic in . We can answer this question exactly for some graphs, specifically those with a perfect matching and maximum degree at least , where is the number of vertices:
Proposition 2.3**.**
Suppose , has a matching of size , and has a vertex of degree at least . Letting , we have and .
Proof.
The upper bound comes from Theorem 2.1. The upper bound is trivial.
The lower bound is obtained as follows. Suppose we have a coloring of where every copy of has at most repeats. If there is either a monochromatic matching or a monochromatic star with edges, then there is a copy of with at least repeats. For a given color, the endpoints of a maximal matching form a vertex cover of size less than , each vertex of which is incident to fewer than edges. Thus each color class has size less than , and so there are at least colors. ∎
Note that the proof of Proposition 2.3 relies on only two forbidden substructures, both of which are monochromatic. The more advanced proof of Pohoata and Sheffer [11] uses two forbidden substructures, but only one is monochromatic. The other contains many colors and is crucial to obtaining lower bounds of the form for fractional (for example, in Proposition 3.5 below).
Proposition 2.3 clearly generalizes to the following statement.
Proposition 2.4**.**
If has a matching of size and a vertex of degree at least , then , or equivalently, .
One may hope that the quadratic threshold for is always the one given in Proposition 2.3, , even when the hypotheses are not satisfied. In fact, this is the case for paths (see Theorem 1.1). However, there are three extreme cases where the quadratic threshold does not even exist: a star with edges, denoted , a matching with edges, denoted , and the triangle. In the following it is understood that . Recall that is the minimum number of colors needed so that every copy of is ‘rainbow.’
Coloring 2.5**.**
Label the vertices and color the edge between and with color if . This shows .
In fact, if we color the edge between vertex and vertex with color , this gives a coloring showing that . To see that , note that every color class must either be a star or a triangle. The number of edges covered by stars is at most , so the number of edges covered by of these color classes is at most .
Coloring 2.6**.**
The edge set of can be partitioned into matchings of size (when is even this partition is known as a -factorization). Color the edges of according to which matching contains them. This coloring contains either or colors, and shows that .
This coloring actually shows that , since each color class must be a matching, and in a -factorization all the color classes attain their maximum size. Similarly, we have .
Corollary 2.7**.**
If for some , then is either a matching, a star, or a triangle. In that case, .
Proof.
If has a matching of size two and a vertex of degree at least two, then by Proposition 2.4. The only graphs without those properties are matchings, stars, and the triangle. ∎
3 Paths
We now turn our attention to the asymptotics of , where is a path on vertices. (Recall that the asymptotics of were found in Section 2.) We show the range of for which is linear in in Proposition 3.1, and the range for which it is quadratic in in Theorem 3.3. The only gap in these ranges is bounded in Proposition 3.5. See Figure 1 for a summary of these results.
Proposition 3.1**.**
For every , we have .
Proof.
[TABLE]
The lower bound follows by Theorem 2.2 and the upper bound by Coloring 2.5. ∎
Now we find the quadratic threshold for paths. We first give a lemma which facilitates the proofs of Theorem 3.3 and Proposition 3.5.
Lemma 3.2**.**
Fix . In any coloring of , one of three things must occur:
- •
there exists a copy of with at least repeats,
- •
* colors are used, or*
- •
there exist at least color classes each with a matching of at least edges, for some .
Proof.
Let be colored with colors from such that every copy of repeats at most colors. First note that every color class has size less than , or else we would find a monochromatic (see e.g. [7]). For a color , let denote the set of edges of color .
Suppose that for every , there are at most color classes each of size at least . Let , and let . Then
[TABLE]
which implies , so , as desired.
Now suppose that there is some such that ; fix one such . Let be the set of colors such that there is a monochromatic matching in color with at least edges. If , then we are done. Otherwise, we have , and we will find a copy of with at least repeats with the use of an auxiliary bipartite graph.
Let , and let be a minimum vertex cover for the edges of color . Since the endpoints of the edges of a maximal matching form a vertex cover and has a maximal matching of size at most , we have . Since the vertices of of -degree less than cover at most edges of , then at least half of is incident to vertices of -degree at least .
Let be the bipartite graph with parts and and an edge between and if the edge is colored with . Note that for a given color , the number of edges incident to vertices of the form is at least . This implies
[TABLE]
On the other hand, it is clear from the definition of that .
We delete vertices of degree at most repeatedly from and until we cannot delete any more, leaving us with a bipartite graph with minimum degree greater than . To show that is not empty, suppose we delete edges from ; observe that
[TABLE]
which can be seen by splitting into the cases and . Thus is not empty, and the minimum degree of is at least .
We greedily construct a path in as follows: select not equal to any of or , and select not equal to any of or . When selecting , only vertices of have been forbidden, so we may choose greedily. When selecting , it is possible that we have forbidden many vertices of , since many vertices of correspond to the same . However, the vertex is adjacent to only if the edge has color , and so has at most one neighbor in for every choice of . Thus at most of ’s neighbors in are forbidden, so we may choose greedily.
If is even, this allows us to construct the path , which has repeats. If is odd, we construct the path , which has repeats (see Figure 2).
∎
We can think of the argument with the auxiliary bipartite graph as a kind of Turán problem: the auxiliary graph is reflective of a hypergraph on (the edges are exactly those -neighborhoods of ), and we wish to find a kind of ‘self-avoiding’ Berge-path in this hypergraph given that the number of hyperedges present is large.
With Lemma 3.2, we find the quadratic threshold for paths with an even number of vertices in the following theorem.
Theorem 3.3**.**
For every , we have .
Proof.
It suffices to show . Let be colored with colors from such that every copy of repeats at most colors. The first case of Lemma 3.2 does not occur, and in the second case we are done. If the third case of Lemma 3.2 occurs, then we have a monochromatic matching of size , with which we have a copy of with repeats (see Figure 3). So this case is impossible as well.
∎
There is a small gap in what we have shown when is odd, namely has only been shown to be (and trivially). Note that is equivalent to the star , so by Corollary 2.7, . The case was shown to be linear by a construction of Rosta [12]: label the vertices of with distinct binary strings of length , and color the edge between vertices labeled and with color (where is bitwise exclusive-or). This coloring is proper, since implies . In this coloring, if any with vertices (in that order) contains only two colors, it must be that and . Upon ‘adding’ these equations, we get , meaning , contradicting this being a path. Thus Rosta’s construction shows .
We only have loose bounds when . First we need a combinatorial lemma of Erdős [4], which we state here for a particular case.
Lemma 3.4**.**
Let be subsets of , where and for some constant . If , then there are some such that .
Proposition 3.5**.**
For odd , we have . For odd , we have , and for , we have .
Proof.
The upper bound comes from Theorem 2.1.
Let be colored with colors from so that every contains at least colors (and at most repeated colors). By Lemma 3.2, we either have a copy of with at least repeats (which does not occur), we use colors (so we are done), or there exists a such that there are at least color classes each with a matching of at least edges. Let be the set of these colors. If , then there are already at least many colors. Otherwise, . For , there is a matching of size in color ; let be the matched vertices of . If an edge of color appears between vertices of but is not in the matching , then we have a with repeats, which is not allowed. If there are five edges of color among the vertices of , these edges must be incident to at least three edges of , and so we may find a with repeats as well (Figure 4 enumerates these possibilities and shows the desired in each case). Therefore there are at least colors (even just among the vertices of ).
To show the lower bound when , again apply Lemma 3.2: the first case does not occur, and in the second case we are done, so suppose we are in the third case as before. If we have many colors in and we are done. So suppose . Let , let be a maximum matching in color , and let . Note that and , the latter of which follows because , since no color class has size more than . We have by Lemma 3.4 that there are two indices and such that . The graph whose edges are the edges of and incident to at least one vertex of contains a disjoint union of even cycles and paths of length at least two on at least vertices. Thus we can find disjoint copies of an edge of color incident with an edge of color , as in Figure 5. Stringing these together into a path with vertices gives at least many repeated colors (with equality when , 11, and 13), which is not allowed.
∎
4 Acknowledgements
This research project was conducted at 2018 CUNY Combinatorics REU. Many thanks go to Adam Sheffer, who ran the program, fostered discussions, and had several helpful comments on this paper. Additional thanks go to a referee for suggestions which improved the clarity of the proofs.
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