MONOCHROMATIC LOOSE PATHS IN MULTICOLORED k-uniform CLIQUES

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1 MONOCHROMATIC LOOSE PATHS IN MULTICOLORED k-uniform CLIQUES ANDRZEJ DUDEK AND ANDRZEJ RUCIŃSKI Abstract. For positive integers k and, a k-uniform hypergraph is caed a oose path of ength, and denoted by P k), if it consists of edges e,..., e such that e i e j = if i j = and e i e j = if i j 2. In other words, each pair of consecutive edges intersects on a singe vertex, whie a other pairs are disjoint. Let RP k) ; r) be the minimum integer n such that every r-edge-cooring of the compete k-uniform hypergraph K n k) yieds a monochromatic copy of P k). In this paper we are mosty interested in constructive upper bounds on RP k) ; r), meaning that on the cost of possiby enarging the order of the compete hypergraph, we woud ike to efficienty find a monochromatic copy of P k) in every cooring. In particuar, we show that there is a constant c > 0 such that for a k 2, 3, 2 r k, and n k + )r + nr)), there is an agorithm such that for every r-edge-cooring of the edges of K n k), it finds a monochromatic copy of P k) in time at most cn k. We aso prove a non-constructive upper bound RP k) ; r) k )r.. Introduction For positive integers k 2 and 0, a k-uniform hypergraph is caed a oose path of ength, and denoted by P k), if its vertex set is {v, v 2,..., v k )+ } and the edge set is {e i = {v i )k )+q : q k}, i =,..., }, that is, for 2, each pair of consecutive edges intersects on a singe vertex see Figure ), whie for = 0 and = it is, respectivey, a singe vertex and an edge. For k = 2 the oose path P 2) is just a graph) path on + vertices. Let H be a k-uniform hypergraph and r 2 be an integer. The muticoor Ramsey number RH; r) is the minimum n such that every r-edge-cooring of the compete k-uniform hypergraph K n k) yieds a monochromatic copy of H... Known resuts for graphs. For graphs, determining the Ramsey number RP 2), r) is a we-known probem that attracted a ot of attention. It was shown by Gerencsér and Gyárfás [3] that RP 2) 3 +, 2) =. 2 For three coors Figaj and Luczak [0] proved that RP 2), 3) 2. Soon after, Gyárfás, Ruszinkó, Sárközy, and Szemerédi [5, 6] determined this number exacty, showing that The first author was supported in part by Simons Foundation Grant # The second author was supported in part by the Poish NSC grant 204/5/B/ST/0688. An extended abstract of this paper appears in [7].

2 2 ANDRZEJ DUDEK AND ANDRZEJ RUCIŃSKI Figure. A 4-uniform oose path P 4) 3. for a sufficienty arge RP 2), 3) = { 2 + for even, 2 for odd, ) as conjectured earier by Faudree and Schep [9]. For r 4 much ess is known. A ceebrated Turán-type resut of Erdős and Gaai [8] impies that RP 2), r) r. 2) Recenty, this was sighty improved by Sárközy [26] and, subsequenty, by Davies, Jenssen and Roberts [4] who showed that for a sufficienty arge, RP 2) ; r) r /4) + ). 3).2. Known resuts for hypergraphs. Let us first reca what is known about RP k), r) for k 3. For two coors, Gyárfás and Raeisi [4] considered ony paths of ength = 2, 3, 4 and proved that RP k) 2, 2) = 2k, RP k) 3, 2) = 3k, and RP k) 4, 2) = 4k 2. Later, for k = 3 or k 8, and 3, Omidi and Shahsiah [22, 23] determined this number competey: RP k) +, 2) = k ) +, 2 and conjectured that the above formua is aso vaid for k = 4, 5, 6, 7. For an arbitrary number of coors there are ony few resuts and mainy for very short paths. For = 2 and k = 3 so caed bows), Axenovich, Gyárfás, Liu, and Mubayi [] determined the vaue of RP 3) 2, r) for an infinite subsequence of integers r incuding 2 r 0) and for r they showed that RP 3) 2, r) 6r. For k 4, and arge r, the Ramsey number RP k) 2, r) can be easiy upper bounded by a standard appication of Turán numbers by counting the average number of edges per coor). Indeed, it was proved by Frank [] that ex k n; P k) 2 ) = n 2 k 2) for n sufficienty arge, from which it foows quicky that RP k) 2, r) kk )r. For 3, a simiar approach via Turán numbers ex k n; P k) ), determined for arge n by Füredi, Jiang and Seiver in [2], yieds for arge r, and, sighty better for = 3, RP k) ; r) kr/2, 4) RP k) 3 ; r) kr. 5)

3 MONOCHROMATIC LOOSE PATHS IN MULTICOLORED k-uniform CLIQUES 3 In) the smaest instance k = = 3, owing to the vaidity of formua ex 3 n; P 3) 3 ) = for a n 8 see [7]), the above bound hods for a r 3: n 2 RP 3) 3, r) 3r. 6) Recenty, Luczak and Pocyn twice improved 6) significanty. First, in [9], they showed that RP 3) 3, r) 2r + O r), then, in [20], they broke the barrier of 2r by proving the bound RP 3) 3, r) <.98r, both resuts for arge r. This sti seems to be far from the true vaue which is conjectured to be equa to r + 6, the current best ower bound. In a series of papers Jackowska, Pocyn, and Ruciński [7], [24], [25]) confirmed this conjecture for r 0. Finay, for = 3, k arbitrary, and r arge, Luczak, Pocyn, and Ruciński [2] showed an upper bound RP k) 3, r) 250r which is independent of k. In the next section we show a genera upper bound, obtained iterativey for a k 2, starting from the Erdős-Gaai bound 2) RP 2), r) r. Theorem.. For a k 2, 3, and r 2 we have RP k) ; r) k )r. Theorem. can be easiy improved for r 3 provided is arge. Using ) instead of 2), we obtain for three coors that RP k) ; 3) 3k 4), and for r 4, by 3), RP k) ; r) k )r /4. On the other hand, for arge r, the bound 4) is roughy twice better than the one in Theorem...3. Constructive bounds. In this paper we are mosty interested in constructive bounds which means that on the cost of possiby enarging the order of the compete hypergraph, we woud ike to efficienty find a monochromatic copy of a target hypergraph F in every cooring. Ceary, by examining a copies of F in K n k) for n RF ; r), we can aways find a monochromatic one in time On V F ) ). Hence, we are interested in compexity not depending on F, preferaby On k ). Given a k-graph F, a constant c > 0 and integers r and n, we say that a property RF, r, c, n) hods if there is an agorithm such that for every r-edge-cooring of the edges of K n k), it finds a monochromatic copy of F in time at most cn k. For graphs, a constructive resut of this type can be deduced from the proof of Lemma 3.5 in Dudek and Pra at [6]. Theorem.2 [6]). There is a constant c > 0 such that for a 3, r 2, and n 2 r+, property RP 2), r, c, n) hods. Our goa is to obtain simiar constructive resuts for oose hyperpaths. In Section 2, we show that, by repacing the Erdős-Gaai bound 2) with the assumption on n given in Theorem.2, the proof of Theorem. can be easiy adapted to yied a constructive resut. Theorem.3. There is a constant c > 0 such that for a k 2, 3, r 2, and n 2 r+ + k 2)r, property RP k), r, c, n) hods.

4 4 ANDRZEJ DUDEK AND ANDRZEJ RUCIŃSKI Our second constructive bound vaid ony for r k) utiizes a more sophisticated agorithm. Theorem.4. There is a constant c > 0 such that for a k 2, 3, 2 r k, and n k + )r + + n )) + r 2 k) k r+ k r+, property RP, r, c, n) hods. For r = 2, the bound on n can be improved to n 2k 2) + k. Note that for r = 2 the ower bound on n in Theorem.4 is very cose to that in Theorem.. For r = k the bound in Theorem.4 assumes a simpe form n k 2 + )2 + nk ). Furthermore, when r k, one can show see Caim 4.2) that k r + + n + r 2 ) n + r ) k r + k r yieding the foowing coroary. Coroary.5. There is a constant c > 0 such that for a k 3, 3, 2 r k, and n k + )r + n )) + r k) k r, property RP, r, c, n) hods. We can further repace the ower bound on n in Coroary.5 by sighty weaker but simper) n k + )r + n r). Observe that in severa instances the ower bound in Theorem.4 and aso in Coroary.5) is significanty better that means smaer) than the one in Theorem.3 for exampe for arge k and k/2 r k). On the other hand, for some instances bounds in Theorems.3 and.4 are basicay the same. For exampe, for fixed r, arge k and k the ower bound is kr + ok). This aso matches the bound from Theorem.. 2. Proof of Theorems. and.3 For competeness, we begin with proving bounds 4)-6). Reca that for a given k-uniform hypergraph H, the Turán number, ex k n; H), is the maximum number of edges in an n vertex k-uniform hypergraph with no copy of H. Proposition 2.. For a k 3 and 3, inequaities 4) and 5) hod for arge r, whie inequaity 5) hods for a r. Proof. It has been proved in [2] and [8] that for a k 3 and 3, except for k = = 3 but see Acknowedgements in [7]), and for n sufficienty arge ) ) ) ex k n; P k) n n t n k t ) = + δ, k k k 2 where δ = 0 if is odd and, otherwise, δ =, whie t = +. Regardess of the 2 parity of, for every ε > 0 and sufficienty arge n, this Turán number is smaer than + ε)tn k /k )!. With some foresight, we require that ε 2 ). Thus, for

5 MONOCHROMATIC LOOSE PATHS IN MULTICOLORED k-uniform CLIQUES 2 5 Figure 2. Proving Lemma 2.2. fixed k 2 and 3 and a sufficienty arge r, the average number of edges per coor in an r-cooring of the compete k-graph K n k) with n kr/2 is n k) nk n k )nk ε) ε) + ε) > ex k n; P k) ), r rk! 2k )! 2k )! which proves 4). For = 3, the formua for ex k n; P k) ) simpifies to ex k n; P k) 3 ) = n k ) and we have n ) k) n r k aready for n kr. Since the ony extrema k-graph in this case is the fu star and it is impossibe that a coors are stars, we get 5). Finay, for = k = 3 it was proved in [7] that ex 3 n; P 3) 3 ) = ) n 2 for a n 8 and the same argument as above appies to a r 3. Preparing for the proof of Theorem., reca that Erdős and Gaai [8] showed that the Turán number for a graph path P 2) satisfies the bound ex 2 n; P 2) ) )n. 2 This immediatey yieds, by the same argument as in the above proof, that the majority coor in K r contains a copy of P 2), and consequenty RP 2) ; r) r. We are going to use this resut by bowing up the edges of a graph to obtain a 3-graph, then bowing the edges of a 3-graph to obtain a 4-graph, and so on. Formay, we ca an edge of a hypergraph sefish if it contains a vertex of degree one, that is, a vertex which beongs excusivey to this edge. We ca a hypergraph H sefish if every edge of H is sefish. Ceary, for k 3 and, the oose path P k) is sefish. A sefish k-graph H can be reduced to a k )-graph G H by removing one vertex of degree one from each edge of H. Inversey, every k )-graph G can be turned into a sefish k-graph H, caed a sefish extension of G, such that G = G H, by adding EG) vertices, one to each edge of G. Note that EH) = EG H ). Lemma 2.2. Let H be a sefish k-graph with G = G H. Then RH; r) RG; r) + r EH) ) +. Proof. Let n = RG; r) + r EH) ) +, V = U W, U W =, V = n, U = RG; r), and W = r EH) ) +. Consider an r-cooring of the edges of K n k). For every k )-tupe e of vertices in U, we choose the most frequent coor on a the k-tupes e {w}, w W see Figure 2). This induces an r-cooring of the edges of the cique K k ) U on vertex set U. By the definition of RG; r), this yieds a

6 6 ANDRZEJ DUDEK AND ANDRZEJ RUCIŃSKI monochromatic say, red) copy G of G in the induced cooring). Note that for each edge e of G the red coor appears on at east EH) k-tupes containing e and, thus, one can find a red sefish extension H of G which is isomorphic to H. Proof of Theorem.. We use induction on k. For k = 2 the theorem coincides with the Erdős-Gaai resut 2). Assume that for some k 3 we have RP k ) ; r) k 2)r, observe that P k ) = G k) P, and appy Lemma 2.2 obtaining RP k) ; r) RP k ) ; r) + r ) + k 2)r + r = k )r. Proof of Theorem.3. To get the desired ower bound on n it suffices to repace in the base step of induction the Erdős-Gaai bound 2) by the one from Theorem.2 which yieds RP k) ; r) RP k ) ; r) + r ) + 2 r+ + k 3)r + r = 2 r+ + k 2)r. It remains to show that the performance time does not exceed cn k for some c > 0, which, by Theorem.2, is the case when k = 2. W..o.g., assume that c. Suppose that for some k 3 it hods for k )-uniform hypergraphs. Simiary as in Lemma 2.2, we arbitrariy partition V = U W with U 2 r+ + k 3)r and W = r ) +. Next we coor each k )-tupe e in U by the most frequent coor on the k-tupes e {w}, w W. This requires no more than ) U W n k /k )! k steps. Finay, by inductive assumption, in time at most cn k we find a monochromatic copy of P k ) in U which can be extended to a monochromatic P k) in no more than W r 2 2 r 2 2 r 2 r n) 2 = 2 r 3 n 2 steps. Atogether, the performance time, using bounds r 2, k 3, 3, and so n 30, is as required. n k /k )! + cn k + 2 r 3 n 2 /2 + c/30 + /960)n k cn k, 3. Proof of Theorem.4 The proof is based on the depth first search DFS) agorithm. Such approach for graphs was first successfuy appied by Ben-Eiezer, Kriveevich and Sudakov [2, 3] and for Ramsey-type probems by Dudek and Pra at [5, 6]. Given integers k and 2 m k, and disjoint sets of vertices W,..., W m, V m, an m-partite compete k-graph K k) W,..., W m, V m ) consists of a k-tupes of vertices with exacty one eement in each W i, i =,..., m, and k m + eements in V m. Note that if W i, i =,..., m, and V m k m)+ for m k or V m for m = k), then K k) W,..., W m, V m ) contains P k) we inductivey find a copy of P k). Indeed, if m k, then in K k) W,..., W m, V m ), edge by edge, by making

7 MONOCHROMATIC LOOSE PATHS IN MULTICOLORED k-uniform CLIQUES 3 7 sure that for each edge e, e W i = for i =,..., m ) and e V m = m k + and the consecutive edges of P k) intersect in V m. In the remaining case, when m = k, the consecutive edges of P k) intersect either in W or V k by aternating between these two sets. We now give a description of the agorithm. As an input there is an r-cooring of the edges of the compete k-graph K n k). The agorithm consists of r impementations of DFS subroutine, each round exporing the edges of one coor ony and either finding a monochromatic copy of P k) or decreasing the number of coors present on a arge subset of vertices, unti after the r )st round we end up with a monochromatic compete r-partite subgraph, arge enough to contain a copy of P k). During the ith round, whie trying to buid a copy of the path P k) in the ith coor, the agorithm seects a subset W i,i from a set of sti avaiabe vertices V i V and, by the end of the round, creates trash bins S i and T i. The search for P k) is reaized by a DFS process which maintains a working path P in the form of a sequence of vertices) whose endpoints the first or the ast k vertices on the sequence) are either extended to a onger path or otherwise put into W i,i. The round is terminated whenever P becomes a copy of P k) or the size of W i,i reaches certain threshod, whatever comes first. In the atter case we set S i = V P ). To better depict the extension process, we introduce the foowing terminoogy. An edge of P k) is caed pendant if it contains at most one vertex of degree two. The vertices of degree one, beonging to the pendant edges of P k) are caed pendant. In particuar, in P k) a its k vertices are pendant. For convenience, the unique vertex of the path P k) 0 is aso considered to be pendant. Observe that for t 0, to extend a copy P of P k) t to a copy of P k) t+ one needs to add a new edge which shares exacty one vertex with P and that vertex has to be pendant in P. Our agorithm may aso come across a situation when P =, that is, P has no vertices at a. Then by an extension of P we mean any edge whatsoever. The sets W i,i have a doube subscript, because they are updated in the ater rounds to W i,i+, W i,i+2, and so on, unti at the end of the r )st round uness a monochromatic has been found) one obtains sets W i := W i,r, i =,..., r, a fina trash set T = r i= T i r i= S i and the remainder set V r = V \ r i= W i T ) such that a k-tupes of vertices in K k) W,..., W r, V r ) are of coor r. As an input of the ith round we take sets W j,i, j =,..., i, and V i, inherited from the previous round, and rename them to W j,i, j =,..., i, and V i. We aso set T i = and P =, and update a these sets dynamicay unti the round ends. Now come the detais. For i r, et P k) and { i ) k r+ τ i = ) + if i r 2, k r+2 k i r 2) if i = r, k r+ t i = τ i + 2i ). 7)

8 8 ANDRZEJ DUDEK AND ANDRZEJ RUCIŃSKI Note that τ i is generay not an integer. It can be easiy shown see Caim 4.) that for a 2 r k and i r τ i i ) + ) k r + + n + r 2 )). 8) k r + Before giving a genera description of the ith round, we dea separatey with the st and 2nd round. Round Set V = V, W, =, and P =. Seect an arbitrary edge e of coor one say, red), add its vertices to P in any order), reset V := V \ e, and try to extend P to a red copy of P k) 2. If successfu, we appropriatey enarge P, diminish V, and try to further extend P to a red copy of P k) 3. This procedure is repeated unti finay we either find a red copy of P k) or, otherwise, end up with a red copy P of P k) t, for some t, which cannot be extended any more. In the atter case we shorten P by moving a its pendant vertices to W, and try to extend the remaining red path again. When t 2, the new path has t 2 edges. If t = 2, P becomes a singe vertex path P k) 0, whie if t =, it becomes empty. Let us first consider the simpest but instructive case r = 2 in which ony one round is performed. We terminate Round as soon as W,. 9) If at some point P = and cannot be extended which means there are no red edges within V ), but 9) fais to hod, then we move W, arbitrary vertices from V = V \ W, to W, and stop. At that moment, no edge of K k) W,, V ) is red so, a of them must be, say, bue). Moreover, since the size of W, increases by increments of at most 2k ), we have and, consequenty, W, + 2k ), V = n W, V P ) n 2k ) + V P k) ) k 2) + by our bound on n see Theorem.4, case r = 2). This means that the competey bue copy of K k) W,, V ) is arge enough to contain a copy of P k). When r 3, there are sti more rounds ahead during which the set W, wi be cut down, so we need to ensure it is arge enough to survive the entire process. We terminate Round as soon as W, k )τ ) If at some point P = and cannot be extended and 0) fais to hod, we move k )τ W, arbitrary vertices from V = V \ W, to W, and stop. Since the size of W, increases by increments of at most 2k ) and the R-H-S of 0) is not necessariy integer, we aso have W, k )τ k ). )

9 MONOCHROMATIC LOOSE PATHS IN MULTICOLORED k-uniform CLIQUES 4 9 Finay, we set S := P, T = for mere convenience, and V := V \ W, S T ). Note that S V P k) ) = )k ) +. Aso, it is important to reaize that no edge of K k) W,, V ) is coored red. Round 2 We begin with resetting W,2 := W, and V 2 := V, and setting P :=, W 2,2 =, and T 2 :=. In this round ony the edges of coor two say, bue) beonging to K k) W,2, V 2 ) are considered. Let us denote the set of these edges by E 2. We choose an arbitrary edge e E 2, set P = e, and try to extend P to a copy of P k) 2 in E 2 but ony in such a way that the vertex of e beonging to W,2 remains of degree one on the path. Then, we try to extend P to a copy of P k) 3 in E 2, etc., aways making sure that the vertices in W,2 are of degree one. Eventuay, either we find a bue copy of P k) or end up with a bue copy P of P k) t, for some t, which cannot be further extended. We move the pendant vertices of P beonging to W,2 to the trash set T 2, whie the remaining pendant vertices of P go to W 2,2. Then we try to extend the shortened path again. By moving the pendant vertices of P in W,2 to T 2 we make sure that in the next iterations there wi be no bue edge e with exacty one vertex in W,2, one vertex in W 2,2 and k 2) vertices in V 2 \ W 2,2. We terminate Round 2 as soon as W 2,2 k 2)τ 2. If at some point P = and cannot be extended and W 2,2 < k 2)τ 2, then we move k 2)τ 2 W 2,2 arbitrary vertices from V 2 to W 2,2 and stop. Note that at the end of this round W 2,2 k 2)τ 2 + 2k 2). 2) We set S 2 := V P ) and V 2 := V \ W,2 W 2,2 S S 2 T 2 ). Observe that no edge of K k) W,2, W 2,2, V 2 ) is red or bue. We wi now show that T 2 t 2 and W,2 k 2)τ 2. 3) First observe that the size of W, the set obtained in Round ) satisfies W, W,2 + T ) Indeed, at the end of this round W, is the union of W,2 T 2 and the vertices in V P ) W,2 that were moved to S 2. Since V P ) W,2, 4) hods. Aso note that each vertex in T 2 can be matched with a set of k 2 or k vertices in W 2,2, and a these sets are disjoint. Consequenty, Inequaity 5) immediatey impies that Furthermore, W 2,2 k 2) T 2. 5) T 2 5) k 2 W 2,2 2) τ = t 2. k )τ ) W, 4) W,2 + T 2 + W,2 + τ 2 + +, competing the proof of 3).

10 0 ANDRZEJ DUDEK AND ANDRZEJ RUCIŃSKI Figure 3. Appying the agorithm to a 7-uniform hypergraph. Here i = 4 and path P, which consists of edges e, e 2, and e 3, cannot be extended. Therefore, the vertices in V P ) W,4 W 2,4 W 3,4 ) are moved to the trash bin T 4 and the pendant vertices in V 4 e e 3 ) are moved to W 4,4. From now on we proceed inductivey. Assume that i 3 and we have just finished round i constructing so far, for each j i, sets S j, T j, and W j,i, satisfying S i V P k) ), and T i t i, and the residua set W j,i k i + i 2 τ i, 6) i V i = V \ W j,i S j T j ) j= such that K k) W,i,..., W i,i, V i ) contains no edge of coor, 2,..., or i. Round i, 3 i r We begin the ith round by resetting W,i := W,i,..., W i,i := W i,i, and V i := V i, and setting P :=, W i,i :=, and T i :=. We consider ony edges of coor i in K k) W,i,..., W i,i, V i ). Let us denote the set of such edges by E i. As in the previous steps we are trying to extend the current path P using the edges of E i, but ony in such a way that the vertices from P that are in W,i W i,i have degree one in P and the vertices of degree two in P beong to V i. When an extension is no onger possibe and P, we move the pendant vertices of P beonging to i j= W j,i to the trash set T i, whie the remaining pendant vertices of P go to W i,i see Figure 3). Then we try to extend the shortened path. We terminate the ith round as soon as W i,i k i i τ i. If P = and cannot be extended and W i,i < k iτ i i, then we move k iτ i i W i,i vertices from V i to W i,i and stop. This yieds that W i,i k i i τ i + 2k i). 7)

11 MONOCHROMATIC LOOSE PATHS IN MULTICOLORED k-uniform CLIQUES 5 Simiary as in 4) and 5) notice that for a j i and Thus, k i + i 2 and, since aso we get τ i W j,i W j,i + T i i + 8) T i i k i W i,i τ i + 2i ) = t i. 9) 6) W j,i 8),9) W j,i + τ i i = W j,i + τ i i + + k i + i 2 τ 7) i = k i + i τ i + +, W j,i k i i τ i. 20) Finay we set S i := V P ). Consequenty, when the ith round ends, we have 20) for a j i. We aso have S i V P k) ), T i t i, and V i = V \ i j= W j,i S j T j ) such that K k) W,i,..., W i,i, W i,i, V i ) has no edges of coor, 2,..., or i. In particuar, when the r )st round is finished, we have, for each j r, W j,r k r + r 2 τ r, 2) S r V P k) ) and T r t r. Set W j := W j,r, j =,..., r, and V r := V \ r j= W j S j T j ) and observe that K k) W,..., W r, V r ) has ony edges of coor r. By 2), for each j r W j 2) k r + r 2 τ r 7) =. Now we are going to show that V r k r+) which wi compete the proof as this bound yieds a monochromatic copy of P k) inside K k) W,..., W r, V r ). Actuay for r k it suffices to show that V r k r) +.) First observe that W, + + W r 2,r 2 W + + W r 2 + T + + T r. 22) This is easy to see, since during the process Aso, W i,i W i,r W i,i T i+ T r )). W, ) k )τ 2 + 2k ) + + 8) k ) + ) k r + + n + r 2 )) + 2k ) + + k r +

12 2 ANDRZEJ DUDEK AND ANDRZEJ RUCIŃSKI and, for 2 i r, Since W i,i 7) k i i τ i + 2k i) we have by 22) that 8) k i) + ) k r + + n r k i) = k r/2)r ), i= W W r + T T r + ) k r + + n + r 2 k + )r k r + + n + r 2 )) + 2k i). k r + )) k r/2)r ) k r + + 2k r)r ) + + )) + r 2 k r + + 2k r)r ) + +. As aso S i V P k) ) = k ) ) + for each i r and r V r = V W i + T i + S i ), we finay obtain, using the ower bound on n = V, that i= V r k + )r 2k r)r ) r ) [k ) ) + ] = 2r 3) + r )r 2) + k ) + k r + ) k r + ), since the first three terms in the ast ine are nonnegative. To check the On k ) compexity time, observe that in the worst-case scenario we need to go over a edges coored by the first r coors and no edge is visited more than once. 4. Auxiiary inequaities For the sake of competeness we prove here two straightforward inequaities. Caim 4.. Let 2 r k, i r and { i ) k r+ τ i = ) + if i r 2, k r+2 k i r 2) if i = r. k r+ Then, τ i i ) + ) k r + + n + r 2 )). k r +

13 MONOCHROMATIC LOOSE PATHS IN MULTICOLORED k-uniform CLIQUES 6 3 Proof. It suffices to observe that k i k r + + k r k i k r + + dx k r+ x ) k i = k r + + n k r + ) k k r + + n k r + = k r + + n + r 2 ). k r + Caim 4.2. For a 2 r k we have k r + + n + r 2 ) k r + n + r ). 23) k r Proof. Let fx) = n + x) and observe that f x) =. Hence, fx) is x+ xx+) 2 decreasing for x > 0 and so fx) im x fx) = 0. Consequenty, for x = k r by assumption k r ) we get that k r + n + k r which is equivaent to 23). ) ) k r + = n = n k r References ) ) k k n, k r k r +. M. Axenovich, A. Gyárfás, H. Liu, and D. Mubayi, Muticoor Ramsey numbers for tripe systems, Discrete Math ), I. Ben-Eiezer, M. Kriveevich, and B. Sudakov, Long cyces in subgraphs of pseudo)random directed graphs, J. Graph Theory ), no. 3, , The size Ramsey number of a directed path, J. Combin. Theory Ser. B ), no. 3, E. Davies, M. Jenssen, and B. Roberts, Muticoour Ramsey numbers of paths and even cyces, European J. Combin ), A. Dudek and P. Pra at, An aternative proof of the inearity of the size-ramsey number of paths, Combin. Probab. Comput ), no. 3, , On some Muticoor Ramsey Properties of Random Graphs, SIAM J. Discrete Math ), no. 3, A. Dudek and A. Ruciński, Constructive Ramsey numbers for oose hyperpaths, LATIN 208, to appear in LNCS, Springer, P. Erdős and T. Gaai, On maxima paths and circuits of graphs, Acta Math. Acad. Sci. Hungar 0 959), R. J. Faudree and R. H. Schep, Path Ramsey numbers in muticoorings, J. Combin. Theory Ser. B 9 975), no. 2, A. Figaj and T. Luczak, The Ramsey number for a tripe of ong even cyces, J. Combin. Theory Ser. B ), no. 4, P. Frank, On famiies of finite sets no two of which intersect in a singeton, Bu. Austra. Math. Soc ), no.,

14 4 ANDRZEJ DUDEK AND ANDRZEJ RUCIŃSKI 2. Z. Füredi, T. Jiang, and R. Seiver, Exact soution of the hypergraph Turán probem for k-uniform inear paths, Combinatorica ), no. 3, L. Gerencsér and A. Gyárfás, On Ramsey-type probems, Ann. Univ. Sci. Budapest. Eötvös Sect. Math ), A. Gyárfás and G. Raeisi, The Ramsey number of oose trianges and quadranges in hypergraphs, Eectron. J. Combin ), no. 2, Paper 30, A. Gyárfás, M. Ruszinkó, G. Sárközy, and E. Szemerédi, Three-coor Ramsey numbers for paths, Combinatorica ), no., , Corrigendum: Three-coor Ramsey numbers for paths [Combinatorica ), no., 35 69], Combinatorica ), no. 4, E. Jackowska, J. Pocyn, and A. Ruciński, Turán numbers for 3-uniform inear paths of ength 3, Eectron. J. Combin ), no. 2, Paper 2.30, A. Kostochka, D. Mubayi, and J. Verstraëte, Turán probems and shadows II: Trees, J. Combin. Theory Ser. B ), T. Luczak and J. Pocyn, On the muticoor Ramsey number for 3-paths of ength three, Eectron. J. Combin ), no., Paper.27, , The mutipartite Ramsey number for the 3-path of ength three, Discrete Math ), no. 5, T. Luczak, J. Pocyn, and A. Ruciński, On muticoor Ramsey numbers for oose k-paths of ength three, European J. Combin ), G.R. Omidi and M. Shahsiah, Ramsey numbers of 3-uniform oose paths and oose cyces, J. Combin. Theory Ser. A 2 204), , Diagona Ramsey numbers of oose cyces in uniform hypergraphs, SIAM J. Discrete Math ), no. 3, J. Pocyn, One more Turán number and Ramsey number for the oose 3-uniform path of ength three, Discuss. Math. Graph Theory ), no. 2, J. Pocyn and A. Ruciński, Refined Turán numbers and Ramsey numbers for the oose 3-uniform path of ength three, Discrete Math ), no. 2, G. Sárközy, On the muti-coored Ramsey numbers of paths and even cyces, Eectron. J. Combin ), no. 3, Paper 3.53, 9. Department of Mathematics, Western Michigan University, Kaamazoo, MI, USA E-mai address: andrzej.dudek@wmich.edu Department of Discrete Mathematics, Adam Mickiewicz University, Poznań, Poand E-mai address: rucinski@amu.edu.p