Improved Monochromatic Double Stars in Edge Colorings

We give a slight improvement on the size of a monochromatic double star we can guarantee in every r-coloring of the edges of Kn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_n$$\end{document}.

This implies that if the edges of K n are r-colored, then either all color classes are connected or there is a monochromatic double star with at least n rÀ1 vertices. Let us denote by f(n, r) the size of the largest monochromatic double star we can find in any r-coloring of K n . The previous remark raises the following natural problem. Problem 4 ([4]) Is it true that f ðn; rÞ ! n rÀ1 for all r ! 3? Note that for r ¼ 2 this is not true as seen by a random 2-coloring (see [4] for details). In [4] we proved the following weaker lower bound on f(n, r). For r ! 3 this improved a result of Mubayi [9], who proved that there is a monochromatic subgraph of diameter at most three with at least n rÀ1þ1=r vertices in every r-coloring of K n .
For r ¼ 2 the bound in Theorem 5 is close to best possible (see [4]). However, it is possible that for r ! 3, f(n, r) is larger, maybe even as large as n rÀ1 (Problem 4). Perhaps surprisingly, despite significant research efforts (see e.g. [7,9,10])), as far as we know there has been no improvement on the bound in Theorem 5. In this note we give a slight improvement on this bound.
there is a monochromatic double star with at least nðrþ1ÞþrÀ1 r 2 þ dn vertices in any r-coloring of the edges of K n .
No effort is made to optimize the d improvement since it is probably far from optimal, we just wanted to make the point that Theorem 5 does not give the right asymptotics for f(n, r). The original question whether f ðn; rÞ ! n rÀ1 still remains open. Note that Letzter [7] proved that there is a triple star of size at least n rÀ1 improving a result of Ruszinkó [10] (a triple star is a tree obtained by joining the centers of three disjoint stars by a path of length 2).

Proofs
Proof of Theorem 6 Our starting point is the argument from [4]. Let r ! 3, d ¼ 1 and M ¼ nðrþ1ÞþrÀ1 . Consider an r-colored K n . We will show that there is a monochromatic double star with at least M þ dn vertices. Consider an arbitrary vertex p of K n and we denote by A i the set of vertices which are connected to p with an edge of color i. If a vertex a 2 A i has more than M þ dn À jA i j À 1 edges of color i going to other A j -s, then we have a monochromatic double star (with centers a and p) in color i with M þ dn vertices. Thus we may assume that this is not the case. Let us denote by H i the graph consisting of the edges of color i going out of A i . Consider the r-partite graph G with partite classes A i obtained by the removal of the edges in H i and inside the A i ; 1 i r. Just as in [4] (see (1) in [4]) from the previous remark and from the Cauchy-Schwartz inequality we can get the following lower bound: h Define G(i, j) as the bipartite subgraph of G spanned by ½A i ; A j . Note that there are no edges of color i or j in G(i, j) by definition. Let d(v, H) denote the degree of v in the graph H and let d k ðv; HÞ denote the degree of v in color k in the graph H. For any edge e ¼ xy of color k, x 2 A i ; y 2 A j , we define s ijk ðx; yÞ ¼ d k ðx; GÞ þ d k ðy; Gði; jÞÞ; t ijk ðx; yÞ ¼ d k ðx; Gði; jÞÞ þ d k ðy; GÞ: Thus we have two double stars, S ijk ðx; yÞ of size s ijk ðx; yÞ and T ijk ðx; yÞ of size t ijk ðx; yÞ, of color k in G. We will sum s ijk ðx; yÞ þ t ijk ðx; yÞ and estimate this sum. Again just as in [4] (see page 534) using the Cauchy-Schwartz inequality again and (1) we get : Since altogether we summed the cardinalities of the vertex sets of 2|E(G)| monochromatic double stars, (3) implies that at least one of them has size at least M À 2dn rÀ2 . To improve on this first note that we must have the following upper bound since otherwise the same argument would lead to a monochromatic double star of size at least M þ dn.
To get an improvement in (3) we will need to add back some of the removed edges in [H i . We will make use of the ''defect form'' of the Cauchy-Schwarz inequality (as in [11] or in [6]): if 1 l m and First we show the following stability result: in G apart from a small number of exceptional vertices all vertices have degrees close to the average degree in every color otherwise we are done again. Let dðGÞ denote the average degree in graph G and let d k ðGÞ denote the average degree in graph G in color k. A vertex x 2 A i ; 1 i r is called g -exceptional for some g [ 0 if (at least) one of the following holds: • There exists k 6 ¼ i for which we have • There exist j 6 ¼ i and k 6 ¼ j; i for which we have d k ðx; Gði; jÞÞ À dðGÞ ðr À 1Þðr À 2Þ [ gn: Then we have the following stability claim: Claim 7 Let a ¼ 12d. One of the following two cases holds: (i) G has a monochromatic double star of order at least M þ dn.
(ii) The number of a 1=3 -exceptional vertices in G is less than a 1=3 n.
Proof of Claim 7 Assume that (ii) does not hold, so we have at least a 1=3 n exceptional vertices for which either (5) or (6) holds (for some k or k, j, respectively). Then there are at least l ¼ a 1=3 4 n (for simplicity assume that this is an integer) exceptional vertices x for which always the same, say (5), holds and always in the same direction, say larger (the proof will be similar in the other cases), so for some k For each of these l exceptional vertices x we pick a k x for which (7) holds. Denote by E ¼ fðx; k x Þg the set of these l pairs, so jEj ¼ l ¼ a 1=3 4 n. We will lower bound the sum of squares in line (2) a bit differently, namely for the summation we will use the defect form of the Cauchy-Schwarz inequality instead of the original. We rearrange the summation in (8) On the other hand D is defined by Hence, by (9) and (10), we get Applying the defect form of the Cauchy-Schwarz inequality with l ¼ a 1=3 4 n, we get Thus using (11) we get an extra term when we lower bound the sum of squares in line (2), which leads to an extra ! an 4 (just using 2jEðGÞj n 2 ) increment in the size of the double star in (3). Thus we get a monochromatic double star of size at least so indeed (i) holds (here we used a 4 ! 3d.) If (5) holds for l vertices in the other direction (smaller), the proof is almost identical. Finally, assume that (6) holds for l vertices in the larger direction, i.e. for some k, j d k ðx; Gði; jÞÞ [ dðGÞ ðr À 1Þðr À 2Þ þ a 1=3 n: ð12Þ We proceed similarly as above. For each of these l exceptional vertices x we pick a pair ðk x ; j x Þ for which (12) holds. Denote by E ¼ fðx; k x ; j x Þg the set of these l triples, so jEj ¼ l ¼ a 1=3 4 n. Here we will use the defect form of the Cauchy-Schwarz inequality for the other part of the summation in (2), i.e.
We rearrange the summation in (13) so that we start with the l terms X Hence, by (14) and (15), we get Applying the defect form of the Cauchy-Schwarz inequality with l ¼ a 1=3 4 n, we get Thus using (16) again we get an extra term when we lower bound the sum of squares in line (2), which leads to an extra Let us take a non-exceptional vertex y 2 N G 2 ðx; A 3 Þ (this is possible because of (20)). Since y is non-exceptional we have d 2 ðy; Gð1; 3ÞÞ ! dðGÞ ðr À 1Þðr À 2Þ À a 1=3 n: Consider the color-2 double star with centers x and y and leaf-sets N ðG[H 2 Þ 2 ðx; VðGÞ n A 1 Þ and N G 2 ðy; A 1 Þ. Using the fact that x is non-exceptional, (19) and (21), the size of this double star is at least (using the estimate in (3) again), a contradiction (using our choices for d and a). Indeed, We gave a slight improvement on the size of a monochromatic double star we can guarantee in every r-coloring of the edges of K n . Let us summarize where the improvement came from. Stability is used to show that most of the vertices are a part of monochromatic double stars of approximately the same size that was obtained in [4]. This allows us to start with a typical vertex, get the double star as we had before, but also get the extra edges from x 2 A 1 to A 2 in color 2 which we didn't have before.
However, the original question whether there is a double star of size at least n rÀ1 still remains open. No effort was made to optimize the improvement, we just wanted to make the point that Theorem 5 does not give the right asymptotics for f(n, r).
With a more careful analysis one might be able to improve this further; perhaps even to the right order of magnitude, d ¼ c=r 3 for some constant c. However, our approach, using only the r-partite graph on the A i -s, is not sufficient to prove f ðn; rÞ ! n rÀ1 ; there is an example demonstrating that. Indeed, consider the following random construction (for simplicity we describe it only for r ¼ 3). We have jA i j ¼ ðn À 1Þ=3; 1 i 3 and we have random bipartite graphs between the A i -s with the following densities: in color 1 between A 1 and A 2 and between A 1 and A 3 with density ð1=4 À Þ for some small and between A 2 and A 3 with density ð1=2 þ 2Þ. The other colors are similar. Then it is easy to verify that there is no monochromatic double star of size at least n/2 in the tripartite graph; the point is that any x 2 A 2 and y 2 A 3 will have a large common neighborhood in A 1 in color 1 with high probability which can only be counted once. Thus in order to prove f ðn; rÞ ! n rÀ1 (if it is true), one needs to use the edges inside the A i -s or of course use totally new ideas.