It is well known that in every r‐coloring of the edges of the complete bipartite graph K m , n there is a monochromatic connected component with at least false( m + n false) / r vertices. In this paper we study an extension of this problem by replacing complete bipartite graphs by bipartite graphs of large minimum degree. We conjecture that in every r‐coloring of the edges of an ( X , Y )‐bipartite graph with false| X false| = m , false| Y false| = n , δ ( X , Y ) > false( 1 − 1 / ( r + 1 ) false) n, and δ ( Y , X ) > false( 1 − 1 / false( r + 1 false) false) m, there exists a monochromatic component on at least false( m + n false) / r vertices (as in the complete bipartite graph). If true, the minimum degree condition is sharp (in that both inequalities cannot be made weak when m and n are divisible by r + 1). We prove the conjecture for r = 2 and we prove a weaker bound for all r ≥ 3. As a corollary, we obtain a result about the existence of monochromatic components with at least n/(r − 1) vertices in r‐colored graphs with large minimum degree.

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It is well-known that in every r-coloring of the edges of the complete bipartite graph K n,n there is a monochromatic connected component with at least 2n r vertices. It would be interesting to know whether we can additionally require that this large component be balanced; that is, is it true that in every r-coloring of K n,n there is a monochromatic component that meets both sides in at least n/r vertices?Over forty years ago, Gyárfás and Lehel [12] and independently Faudree and Schelp [7] proved that any 2-colored K n,n contains a monochromatic P n . Very recently, Bucić, Letzter and Sudakov [4] proved that every 3-colored K n,n contains a monochromatic connected matching (a matching whose edges are in the same connected component) of size ⌈n/3⌉. So the answer is strongly "yes" for 1 ≤ r ≤ 3.We provide a short proof of (a non-symmetric version of) the original question for 1 ≤ r ≤ 3; that is, every r-coloring of K m,n has a monochromatic component that meets each side in a 1/r proportion of its part size. Then, somewhat surprisingly, we show that the answer to the question is "no" for all r ≥ 4. For instance, there are 4-colorings of K n,n where the largest balanced monochromatic component has n/5 vertices in both partite classes (instead of n/4). Our constructions are based on lower bounds for the r-color bipartite Ramsey number of P 4 , denoted f (r), which is the smallest integer ℓ such that

Let f (K n , H, q) be the minimum number of colors needed to edge-color K n so that every copy of H is

Gyárfás and Lehel and independently Faudree and Schelp proved that in any 2coloring of the edges of K n,n there exists a monochromatic path on at least 2⌈n/2⌉ vertices, and this is tight. We prove a stability version of this result which holds even if the host graph is not complete; that is, if G is a balanced bipartite graph on 2n vertices with minimum degree at least (3/4 + o(1))n, then in every 2-coloring of the edges of G, either there exists a monochromatic cycle on at least (1 + o(1))n vertices, or the coloring of G is close to an extremal coloring -in which case G has a monochromatic path on at least 2⌈n/2⌉ vertices and a monochromatic cycle on at least 2⌊n/2⌋ vertices. Furthermore, we determine an asymptotically tight bound on the length of a longest monochromatic cycle in a 2-colored balanced bipartite graph on 2n vertices with minimum degree δn for all 0 ≤ δ ≤ 1.