One of the bewildering array of characters that peoples this epic classic, Eurymachus is a character who is one of the suitors that flock around Penelope, and, as the text tells us, the most likely one who will actually gain this prize. His name literally means "wide-fighting."
Even though he is said to be the suitor most likely to be successful in gaining Penelope's hand, he is shown to be a rather unpleasant individual, to put it mildly. He is arrogant, hypocritical, abusive and cowardly. Perhaps because of these list of faults, he is the second suitor to be killed by Odysseus. What is interesting about him is the way that he tries to make a deal with Odysseus to prevent being killed in a fit of cowardice. However, when he offers to pay for the damages that he and the other suitors have caused to his household, Odysseus responds using almost precisely the same words as Achilles uses when Agamemnon offers a ransom for Briseis in Book IX of The Iliad.
Not only is he an arrogant and cowardly individual, therefore, but his offer to try and save his life is deeply distasteful and shows his low character in comparison to the heroic nature of Odysseus. He, like so many of the other characters in this classic, acts as something of a foil to the character of Odysseus.